You are fine with just a passport so long as you are Czech, that is only for countries which joined in the last enlargement. So if you're Bulgarian or Romanian then you will need a visa. You're biggest problem will be the short length you want to stay.
You're English seems to be very good so you shouldn't have a problem with language, at least written (some accents can be hard to understand :P I'm looking at you, Scots). Classes are very cheap and there are a lot of them in London. I have a friend how managed to find one for £10/week.
Flats are difficult to come by and you might have to stay in a hostel a week or so. It might even be easier to live in a hostel, you can get small rooms that you can share with another couple and these generally cost a bit more than renting, but not much if you can find a good deal. Letting Agencies generally don't accept international applications, even those from the EU so you will have to find a private landlord. This isn't too hard but something for just two months isn't going to be that easy.
You might want to wait till June, as students leave the City and there are usually a lot of short-term rooms available. Loot is an excellent site for finding private, legit lenders, I've included the link.
You should check the jobseekers sites, there are often a lot of technical jobs but temp ones are hard to come by. Gumtree is your best bet for general jobs, or the government website. I've included a bunch of links below.
If you are working in Czech then it might be a good idea to apply from the there and fly to the UK for the interview. This can be expensive but its better than living in a hostel for a month without any job prospects. And if you're picky about what jobs you want you might have a long wait :S Either way I would advise you to have about £1,000-£2,000 saved, best to play it safe.
The job market is difficult everywhere so you shouldn't let that deter you, just keep a cool head because it's always tough to find something right, even in good times. If you're willing to take the first thing that comes along then you shouldn't have a problem.
Edit: My finance is a Java Developer from France, she moved to the UK and it took her 3 months to find a job, she did turn down a couple offers because they didn't offer her enough money :) French people [shakes fist]. And when her friends come to visit they bitch about how France is so much better, but buy so much stuff that they have to get another suitcase...happened three times :) Apparently Primark and TKMaxx are shopper Meccas.
http://www.loot.com/
http://www.gumtree.com/
http://www.direct.gov.uk/en/Employment/index.htm http://www.reed.co.uk/
http://www.job-seekers.co.uk/