Strand Palace for the location - don't book a "cosy" room, they really are (double bed pushed against the wall). Finished a major refurb just before Covid hit. I work for a travel company selecting hotels in London to work with, and it was our best selling London hotel last year.
The Tower Hotel for views of the Thames / Tower Bridge. Will have good deals on Secret Escapes or Travelzoo.
Doubletree by Hilton Tower of London - again for the views, is usually quite pricey but without business travel it needs guests!
Clermont Charing Cross - really good Trip reviews, fairly recent refurb. Great location, fab service.
Resident Soho or Resident Covent Garden - seconding Hermione's recommendation. Larger than your average London hotel room, there's a cupboard with a kitchenette so you have a drinking water tap, fridge, microwave, kettle, nespresso machine. Iron and ironing board under the bed. Covent Garden is no 1 on Trip but they all get good ratings. They don't have hotel facilities ie no bar / restaurant but they give you a discount book for money off local places and the front desk staff give recommendations, book for you. Definitely book direct with these guys for the best price, they don't do discount websites.
Park Plaza Westminster Bridge for views of the river and Big Ben. The studio rooms are a really good size and I like how it feels more like a junior suite as the wardrobe splits the sleeping area from the rest of the room.
Those are all 4* and very likely to come in under budget for February.
A 5* I'd recommend is Sea Containers. On the South Bank, views of St Pauls. It's modern, quirky, fab cocktails. Not horrifically expensive like the luxury hotels so may come out on budget for your dates, worth a look.