If your just looking for a room to sleep and shower then it's also worth considering Premier Inns, at the moment they have a
Premier Saver so the room rate is discounted but you need to pre-pay now in full and if you want to cancel or amend then there's no refunds.
But if you need the flexibility to amend then you'll need to book a
Premier Flexible rate and that's possibly over your planned budget.
Here's one example, it's at London County Hall (sorry no A/C) it's on the south side of the Thames and it's only about a half mile walk to Westminster Bridge, once over the bridge you have the Houses of Parliament and then about a mile away there's Buckingham Palace.
Rates are cheaper over the weekend and so long as your children are under 16 you can book a family room for 4.
London County Hall
Price breakdown
Premier Saver
Price per room(inc VAT & taxes)
Fri 17 Jun - £69.00
Sat 18 Jun - £79.00
Sun 19 Jun - £69.00
Total 3 nights - £217.00 (approx $332)
Premier Flexible
Price per room (inc VAT & taxes)
Fri 17 Jun - £132.00
Sat 18 Jun - £152.00
Sun 19 Jun - £109.00
Total 3 nights £393.00 (approx $600)
http://www.premierinn.com/en/london-hotels.html
Good luck searching and also check out Travelodge at,
https://www.travelodge.co.uk/uk/london/hotels-in-london