Neighborhood Suggestions?

Nobody can tell you what's "enough", only you can decide that.

Personally, I think that many (most) people make a major mistake by planning trips to Europe that are too short. Your airfare cost is the same whether it gets you 4 days in Europe or 40. Unless you have unlimited money to spend on flights, and can go as often as you want, then it makes (financial) sense to stay longer to get more trip for your flight money.

For me (and I think for many people who may or may not realize it), I don't think it's worth it to go to Europe for less that about 18-21 days. My trips are much better if they're 3 weeks or more - it takes 2 days to get there and another day or so to recover from the travel. Of course, you gotta do what you gotta do - some folks can not imagine taking more than a week away from home, and work demands limit time away for many people. IME, between 3 and 4 weeks is the ideal trip length - but only you can say what's right for you.

Is ANY time ever enough? I think with 15 days (assuming that does not include travel days to/from the UK), you have made a wonderful start to your time in England & Scotland. My idea is that IF you can take enough time to spend another few days there...then do so. As the previous poster said, you've already paid for your plane fare. Make the most of that.

We are frequently in the three to six weeks but longer trips will almost always include a cruise in the middle. Our shortest trip was five days to attend a birthday party. You can make any amount of time work if you adopt the attitude that "we will see that the next time." And there is always a next time.

I hope you have a great time. I've been to London several times, once I was living and working there for 2 months, and it looks like my next trip to London will be in April for 15 days. Since you have other trips planned, it might be a good idea to get that extra week if you can. I'm finding great prices at Travelodge.co.uk for April, their web site allows you to search for an entire month, and then you can find consecutive dates which are the lowest prices in a row, if your dates are flexible. I'd imagine July/August is going to be a whole different thing, but the search process on airbnb and other sites like sabbatical.com (if you have an educational email address) would be similar.

I agree with you on the lower-stress focus, this trip I want to just stay in Paddington for a couple of nights so I can hop the Heathrow Express and drop the whole drama of the hour+ trip on tubes, trains, cabs, etc when I'm exhausted. I found some high ratings and better prices in that area on Booking.com. Europa House Hotel £75/night, so I'll leave a review in the other section, after I go there. You may want to land near wherever you want to travel from, my daughter and her friend took a coach bus from Victoria Station to Bath and Brighton, so that might be a good area to be in for your travels. Have fun!
-Alison

You can search for apartments on booking.com and filter by price. Pick out 5 and check out the neighborhoods closely for convenience and ambiance.

Personally, our favorite neighborhood is Southwark near Tower Bridge, as close to the river as we can get. We love starting each day with a walk along the traffic-free Thames Path, with views across the river to the Tower of London, St. Paul's, and other iconic sights.

The only downside is that it is hard to take a photo upriver without getting that horrible Cheesegrater building.

We always stay at Bayswater/Kensington section. It's close to the Paddington Train Station where the Heathrow Express train arrives from the airport--in 15 minutes. It's also within a short walk of 2-3 of the underground stations, and on the north end of Hyde Park.
If Kensington is okay with Princess Kathryn and Prince William, it's okay with me.

Central London locations are pricey, which is why most people can’t afford to live there. £100 maybe low for July/August.

Areas such as Chiswick, Islington, Richmond or Highgate are slightly further out, but sound as though they might suit your requirements.

We have stayed near St. Paul’s, in Westminster, SOHO, and Mayfair. They were all great! The last trip was two weeks of doing exactly what you plan to do.

The hard part is going to be a vacation rental for the budget you mention. I tried to save some money on a rental last trip and looked outside the center, but then found we would be on a train at least 20 minutes each way into and out of the city every day, so to me, a false economy as lost time and train fares add up. You can probably stay in a Premier In for that price but you will, have to buy all of your meals, so you lose some economy there. IMO, you need to double your budget to stay in close-in in London or look at smaller commuter towns outside and consider the impact of taking the train everyday.