The museums are great. I tend to like modern art so the Guggenheim (beautiful building), the Whitney (classic pieces, I'm not so keen on their new location but it is conveniently located for the High Line) and MOMA (normally does some outstanding exhibitions) would be top of my list. But also worthwhile are the Met, the Frick and the Science museum (the science museum isn't as good as London's, but I think the Met beats the V&A and the British museum for presentation). Agree the Tenement Museum is really fascinating and well worth a visit.
There are a bunch of other museums that seem to be all flash and no substance (like the Museum of Arts and Design on Columbus circle, which is okay-ish, but not as good as you'd expect from the location and facade unless they have a particularly good exhibition running and the museum of Sex which is, frankly, dull), some of the smaller less flashy museums are interesting but they seem to be pretty hit and miss and none were stand outs for me. There is supposed to be a Broadway Museum opening up this year (not sure if Covid has scuppered it) which could be incredible given it's NY.
The Arts in general are incredible, though. The architecture is incredible and you've already got some of that covered, I would encourage a trip around the southern end of the island either by boat or helicopter. And a visit to Grand Central.
Music tends to be great. Find a performance or two. Obviously there's the Lincoln Center and the Met Opera but the scene is so much richer than that and it's worth looking at listings for what's on in smaller spaces. If you like jazz I thoroughly recommend the Village Vanguard.
The food tends to be really good. Obviously there are the super expensive Michelin starred ones (I recommend going at lunch if you really want to try these - less expensive, easier to get into and you won't be as jet lagged) but there are so many good places all over the city. Diners and bistros and pizzerias and one or two phenomenal food trucks. Walking around and choosing at random has its own charm.
Also, swank hotel bars can be fun. An excuse to sip cocktails and pretend you live an entirely different life!
I agree that Central Park is more amazing to New Yorkers than to most Brits, but it can be fun in the summer for people watching, especially on the weekend when the restaurants and shops etc. are more crowded. Also, it's nice to see a bit of greenery if you've been immersing yourself in the buildings of the City for too long!