recommendation for medium price place to stay

What is medium-priced to you? To some it is 300 GBP a night, to others 150.

"Medium price" is nebulous. Please define per night the amount of money you can spend so we can offer specific suggestions. Do keep in mind that exchange rates will fluctuate between November and May.

Have no idea you idea of medium price. But try the Best Western near Victoria Station. We think it is reasonable and very handy to local transportation with nice restaurants in the area.

When describing what medium pricing in London means to you, remember that May has two bank holidays - an early one and a late one - which raise the prices and fill the attractions.

"Assuming" OP's budget for London is similar to her "under 200 euro' she wants for Paris, it looks like "medium" price is ~ 175 GBP

I am not familiar with London at all. I’m guessing that it is quite expensive. I would like to have a room medium priced meaning $200 in that range. Trying to find out what areas I should stay in that will have best access to the iconic things I want to see

Thank you. $200 is about £156 today. That is on the low side but not impossible. Premier Inn is a chain we have used and enjoyed and you can probably stay at one for that amount. There is noe one place to be close to everything London has to offer, but near St. Pancras Station has worked well for us. Others here like the Westminster area. Near St. Paul’s has advantages, too.

You can go to the Premier Inns website and see their location with proximity to major sites. Public trans by bus and tube is great all over central London.

pecanlady,
Confirming your assumption that London is one of the more/most expensive hotel cities. Basically little you can do, it's just the London hotel market. As usual, you get what you pay for. And expect small rooms, compared to US.

While sitting in front your computer use Google Maps satellite view and type in Big Ben in the search engine. As you use the zoom you’ll will see how close many of the iconic sites are.

Parliament, Westminster Abbey, Churchill War Rooms, Horse Guards, Trafalgar Square, Buckingham Palace, and the London Eye are all within easy walking distance of one another.

Also look for the Red Circle with horizontal Blue rectangle running through it which is signage for London Underground.

Public transport via the London Underground can get you around London to other “ iconic sites.

Secondly, type in London Tube map. Simply stay with in Zones 1 & 2.

We stayed at a place called Number 63 a couple of years back, and were very happy with it. It is right across from Hyde Park and very close to the tube.

I’d recommend Luna & Simone. The #24 bus stops right outside their door

How many people? Hub by Premier Inn is an excellent choice for a solo traveler. It might be ok for a very close couple . The rooms are quite small, but modern and clean. Otherwise, I’d still stick with the regular Premier Inn chain or perhaps look at Ibis. Ibis will occasionally have sales, such as pay 2 nights, get the third one free.

Sunday nights in London can be amazingly inexpensive, so that’s one way to save money.

As for unsolicited advice, if you find your lodging soon, I would book it and prepay if you have set dates.

The exchange rate now is fantastic for you perhaps because the Brexit deadline is March 29 and there is a political football now with the EU agreeing to a UK departure agreement but the UK's parliament may not accept it. Will the UK's relationship to the EU be like Norway's? Or like Japan's? Who knows....and perhaps because of that, the $ is strong right now. When May (the month, not the PM) comes along, the £ could likely regain its strength and the "medium" price could be a LOT more expensive and thus you might have to look at hotel options which may not be as nice. Nobody has a crystal ball.

I will say that I just followed the unsolicited advice I provided above. We will be traveling to the UK in mid-to-late March and are THRILLED with the exchange rate of £1 = $1.275 as we just prepaid for our hotel. We'll be returning home just a few days before the March 29 deadline.

The pound could well go lower than $1.27 before March. It was down to $1.20 at one point in 2017, so $1.27 is not as low as it has been recently.

https://www.macrotrends.net/2549/pound-dollar-exchange-rate-historical-chart

I will be watching and will book a non-refundable prepaid hotel for one night in June, but not yet.

1) London has a world class public transportation system in their Tube, Bus and Train system. The system is easy to use, a bargain in price and folks are quite kind in helping you get to your destinations. Please take the time to use YouTube to find videos explaining the ease of purchasing tickets and how to use the system. Some tips:
A) The Oyster Card will be your pass for using the system. Your research will discover many references to the Oyster Card which may create some first time angst; RELAX! The system is easier then it sounds to utilize and you have English speaking folks to help you!
B) The Tube is great, but riding buses is fun and adds more opportunity to site see. Using buses is easy.
C) Walking is highly recommended for the purpose of gaining the slower pace "Ah Ha" moments you will find. Example: by the Victoria and Albert Museum we came across the buildings left unrepaired from the WWII bombings. Seeing this damage left an impression on the devastation Londoners endured during the war.
2) London is an expensive place for accommodations. With four nights in London, proximity to your priority destinations becomes higher at the expense of spending more money on accommodations. However here are some tips for reducing expense:
A) Save money on meals by staying at a BNB or apartment where you can control fixing your own breakfast. Take a deep dive into researching BNB's and room rentals to determine best value for desired location.
B) Many of the landmark destinations have their own dining rooms where we found good value and great food. Eating within these locations helped us sit and relax in the midst of the visit while continuing to enjoy the surroundings. Westminister Abby, Churchills war rooms and Kensington particularly stand out.
C) We shopped in local grocery stores to purchase some fruit and mid-day type snacks to carry in our small backpack. Doing so saved money and placed us in control of having a bite to eat when we needed it.
D) Search for local restaurants located outside of tourist zones where the menu prices do drop off. We had a lot of fun trying pub food.
3) Pre-purchase on line tickets to the top destinations in order to avoid waiting in lines. The most valuable thing you have while on vacation IS YOUR TIME so take every advantage to protect your time.
Enjoy!

I want to 2nd the Luna and Simone recommendation, but add a few comments. I have stayed here a couple of times. The location near Victoria Station is very good. The neighborhood (and Victoria Station) has many restaurants, grocery, liquor store, ATMs, pubs. As mentioned a bus runs right outside the door and even when I went to the theater I felt perfectly safe coming home on the bus alone at night. The rooms are plain and simple and some are very small and there is NO lift. So if you have mobility issues see if they have availability on the lower floors. The breakfasts are filling and supervised usually by one of the brothers whose family has owned and run this hotel for many years. Last time I was there was 2016 and my bathroom had just been renovated and was VERY nice.

I really like Premier Inns when traveling to the UK. This is chain hotel with several properties in good locations in London. The County Hall location is quite popular. These are your basic hotels. Clean, safe, restaurants on site and reasonably priced.
I looked up a price quote for the end of May and came up with 567 pounds for 4 nights. Well within your budget and a good value for the money spent.

Hello pecanlady, I'll just share my own limited experience. In 2015, I had a night before joining a tour in London. The tour hotel would be very expensive for an extra night, so I started looking for a low-budget room in a central location. Many of the rooms I found for around $125 listed the square feet of the room and were really tiny, I remember one was 43 square feet. Yikes I thought. In the end, I raised my budget and found I could get 5 times the room space and quality for about twice the cost, and I had a memorable stay in a lovely, spacious room on the south bank, with a view of the London Eye in the distance. Of course, that was for just one night, not four.

That was our requirements when we went to London in April 2017. We ended up staying in the Pimlico area. It's close to Victoria Station (10 minute walk) and in a very quiet and nice area of London. Lots of restaurants nearby and many small hotels and B+B's. Some are recommended by RS. We ended up staying at the Georgian House Hotel, next door to the Albanian Embassy. It doesn't have a lift (elevator), but it was very comfortable with a great breakfast.

The pound could well go lower than $1.27 before March. It was down to
$1.20 at one point in 2017, so $1.27 is not as low as it has been
recently....I will be watching and will book a non-refundable prepaid
hotel for one night in June, but not yet.

It sure could. I booked at $1.275 and am very happy with that. If the £ weakens further, so much the better for foreign travelers. Good luck with your hotel pre-booking!

London is an expensive place for accommodations. With four nights in
London, proximity to your priority destinations becomes higher at the
expense of spending more money on accommodations.

The time of year one travels can drastically lower the cost of London accommodations. Look online in this 4th quarter of 2018 for hotel deals that are good through March 31 or even through April 30 for example. I now travel early spring because of better pricing of hotels.