If you were still going to work you wouldn't have to tell your employer that you were seeking work elsewhere. The situation doesn't change when you're 'off sick'.
Since it's the Government which actually gets the bill for your SSP (because your employer reclaims it from HMRC) they would be extremely happy for you to cease to 'be a burden to the taxpayer'!
What do you tell a potential new employer? The truth!
("Following a stroke I am now no longer able to continue in my former employment, as I am unable to walk long distances. However my medical condition does not impair my ability to work within an office environment").
If you get a new job you simply resign from your current position. Given that you're not currently working, and in receipt of SSP, neither the employer nor the Government would be likely to be bothered if you asked for a shorter period of notice to apply (if necessary).
There aren't a lot of jobs out there (I've been trying to find one for over 6 years, so I should know!) but I'll wish you luck anyway ;-)
Chris