I just did a quick check at the Premier Inn site and did a dummy booking for 2 adults and 2 kids in mid-July for 5 nights. and came up with around £850. I don't know your dates or if there are 2 adults and 2 kids (must be younger than 16 to qualify for a family room) but it's worth checking out.
The Premier Inn County Hall is a good location. Their family rooms were fully booked on July 16-21 but might be open on your dates. You could also check Premier Inn Waterloo, which is just up the street from County Hall, London PI Paddington, or London PI Victoria. They're all over London. The Premier Inn Holborn is somewhat near Lincoln Suites, as is the Premier Inn Leicester Square.
Premier Inns are a chain hotel based in the UK. I've stayed at a number of them (as have others here on this forum) and it's a good option, especially for families. The beds are comfortable and the rooms are clean. They offer a continental breakfast or cooked breakfast for an additional cost. If you're looking to save money, it's definitely a good option. https://www.premierinn.com/gb/en/home.html
By the way, as others have mentioned, you will get more help if you give us more information. Dates and ages of children would definitely help.