Yes, look at the Premier Inn availability in London. But as also stated, the City of London is not the most desirable location for a leisure traveler, vs. someone who's traveling on business where being in the financial district is a plus. Neighborhoods I've enjoyed include Earl's Court, South Kensington, Pimlico (Luna Simone Hotel, but it's probably booked up for next week), and Bloomsbury, and my current favorite place to stay is a small B&B type hotel on Bayswater Road just across from Hyde Park and a few steps from the Lancaster Gate tube station. (I think it's booked up for next week but check for yourself; it's safe and affordable.)
Depending on your hotel's location, a good option would be to take the Underground (Tube) from Heathrow into London. It's the least expensive option but does require you to handle your own bags and most stations have steps. Although there are ways to minimize or avoid steps/stairs for specific journeys.
London's mass transit (tube, buses, more) is operated by Transport for London (TfL). Get an Oyster card and load it with enough for a week or so traveling. If you're taking the Tube from Heathrow you can buy the Oyster card there, and ask a staffer for help if you need it. The card, if you do the pay as you go (PAYG) thing, will automatically calculate your fares as you use it and once you hit the daily cap for the areas/journeys you're making it will not charge you for additional trips within that cap. I wouldn't mess with travelcards, just load it with PAYG funds. Most of the places that tourists want to visit are located in Zones 1-2 (the central area) of the TfL system. Most hotels are going to be within those fare zones too.
To use the Oyster card for Tube journeys, tap in at the station's entry turnstile, and tap out when you exit the station at the end of your trip. You also use the Oyster card on buses, but just tap in at the card reader when you enter the bus, and do NOT tap again when you exit the bus.
Other transport options from Heathrow into London include a coach (bus) service that goes to Victoria, trains that run to Paddington, taxis and prehired car services (those last two are rather expensive for one person, just IMO).
Once you know where your hotel is, feel free to ask about transport from Heathrow (and back again).
I agree, London Walks, mentioned above, is a brilliant way to see fascinating things in London and learn about what you'e looking at. I've taken more of their walks than I can remember over the years, and have never been disappointed.
London's my favorite city and I hope you have a good time there.