Well the last 2 times I've been in London (and my next visit this summer) I've stayed in the Copthorne Tara just off High Street Kensignton as I've managed to get good deals there - just over €100 a night. The tube stop is a 5 minute walk away, rooms are nice a decent size for London and it's a nice area.
If you were set on Covent Garden, there is a travelodge there (one of the poorest travelodges i've stayed in) but that does tend to attract stag/hen groups so I don't think would be suitable for you.
I have often stayed in a premier inn which is good and clean and well priced. We stay in the Euston area as it is not far from the centre and close to the tube stop. I have also stayed in the City inn in Westminster but would be suitable for a family stay.
I would second the Copthorne Tara, very civilised without being stuffy. I also like the Hilton Metropole on the Edgeware Road, you can get good deals if you book well in advance. Directly across the road from the Tube station. Hotel leisure centre is fab.
Ive stayed in Travel Lodge, Tower Bridge with just me and my child. Great value, opposite a tube stop (free travel for children) but walking distance to many sights, London Eye, Tate, Dungeons etc. Basic, clean, cheap and safe.