Given the location of the accommodation, I'd definitely recommend making use of the nearby parks, which are a lovely change of pace to the hustle and bustle of London.
Walking also strongly recommended, and a cheap A2Z pocket map is still a good investment even in the digital age. So many default to the tube, but often end up walking further in a grungy underground environment, waiting around and feeling hassled, when they could simply have walked in the (relatively) fresh air outside, seeing things en-route. One remarkable thing about London is the sheer density of famous landmarks, and many times I've stumbled across them by accident, simply plotting a route from a to b.
What's nearby? Big Ben and Houses of Parliament for sure, as is the Tate Britain gallery. For food, Fortnum and Mason are famous for their post hampers, but it's also a food store of the 'finest' quality. More humdrum, yet brilliant, is the last remaining Italian deli on Old Compton st, I Camisa. It's an absolute classic unfussy alimentari, and could give you that Italian fix (and they kindly tolerate my attempts to speak Italian - they are all native Italians). There are no fancy signs / fittings, but the food is very good indeed, and very true to Italy.
Soho/Covent garden nearby, and the latter often vibrant in the evenings / weekends
My suggestion would be to choose one day where you do all the stuff you want to see that's in walking distance, finding time to sit down for a coffee, tea or even afternoon tea, to break up the walking.
Then on the other day, either:
- pick a place you absolutely want to go, plot the tube (or bus) journey, and do the same thing of finding everything that appeals in walking distance OR
- Commit to the tube and accepting the grunge for it's ability to get you around the city, and use it to get to see as much as appeals. It's the more normal tourist approach and I can see its appeal, plus for many the tube is an experience in its own right, not the grubby but sometimes necessary public transport I've used over the years.
If you wanted a day trip, Victoria train station is close, so you could hop down to Brighton for the day! It's about an hour each way, with the station in Brighton a 20-25 minute walk from the pier (from memory - I used to live there many years ago)