There's plenty of cheap accommodation in Amsterdam available on this website:
http://www.hostelworld.com/
(Don't be put off by the word 'hostel' in the site's name. Just make an initial search, then use the 'refine options' facility to replace 'hostels' with 'hotels' and 'guesthouses').
For example, the Hotel Paganini will charge you £72.48 per person for a 4 night stay which doesn't include Friday or Saturday nights. 4 nights including both Friday & Saturday would cost £98.72 per person. (There are lots more to choose from).
That just leaves the flights. Here's a complete list of the airlines currently operating to Amsterdam from your selected airports:
Gatwick: BA & easyJet
Heathrow: BA, KLM and bmi
Stansted: easyJet
London City: BA, KLM and Air France (CityJet)
Luton: easyJet
East Midlands: bmiBaby
I've found it easy enough to find inclusive return fares in February, with bmiBaby from East Midlands, of £86.98 per person. Some sample dates with easyJet (from Luton) produced inclusive return fares of £45.98 per person. I've not checked other airlines or departure points, but it seems that you should have little difficulty in finding something suitable.
Chris