1) Go to a show.. Highly recommend BACK TO THE FUTURE THE MUSICAL or COME FROM AWAY as youll both be old enough to get the references.
2) Hotel budgets are a nightmare - literally from £50 to £5000 anywhere in central. Spend some of those crypto gains and go posh! But really there's hotels on almost any street so anything in Zone1 will be fancy. Id love to stay in the hotel in The Shard one day.
3) SkyGarden is free, but you need to prebook a ticket which can be annoying. There's two restaurants up there where you can book for a meal and you get access to the view too. Shard view is great but expensive (£30 each?) An alternative free rooftop is "The Garden at 120 Fenchurch" which is just that - a rooftop garden about 15 stories up. Great views. TOTALLY FREE. There is a posh restaurant on the floor below the roof too if you really want to eat ($$$)
4) You can walk most of Zone1. BukPal, Parliament, Whitehall, Trafalgar Sq are easy walks and if its a decent day actually quite pleasent.
5) Museums are free, but special exhibitions have a ticket charge. Both tates are worth visiting for the free bits, as is National Portrait museum and the Natural History museum
6) For eating - again theres something for every budget - Anything up a skyscraper is a bit pricier so places like SushiSamba and Duck&Waffle carry a premium but are probably worth it for the money for a special one time treat.
7) If youre into Hipster/Tourist markets there's plenty of choice - Portobello, Borough, Broadway are all enjoyable.