I have been looking for a job in England for the past month. I know getting into Europe these days, let alone UK, climbing Everest is easier I guess!

But the main problem I am having, is to know which companies do sponsor a Tier 2 work permits. Most of the job boards I look at dont even mention if a sponsorship is available for a role or not (almost 98% of the jobs listed). I tried applying to the big companies who I saw still sponsor work permits. I got it from a UK gov website which has the list of companies which are eligible for sponsoring work permits. Sorry, I dont have the link handy at the moment to post it here. Most of the companies come back with the question, "Are you eligible to work in UK"? If I did, I wouldn't have mentioned "I need sponsorship" on my resume now, would I? And that is where it all goes downhill. Some which do respond, put in such steep expectations, that I get put off. Like flying to England on a short notice for the interview. In the modern day of Skype, that seems to be a bit weird.
I just wanted to know, are my efforts futile or there is hope. If there is hope, is there a better way to go about my job search? Am I going about it the wrong way?