SSP is available for up to 28 weeks:
https://www.gov.uk/statutory-sick-pay/overview
An employer can terminate the employment of an employee if it becomes apparent that they can no longer do their job (and won't be able to do so for the foreseeable future). There's no fixed time limit. (e.g. if someone who was doing a job that required them to do complex tasks with their hands, and they were to lose both hands in an accident, the employer wouldn't have to wait for any pre-determined period before terminating their employment). However many employers would wait until SSP ends (and possibly beyond) before terminating someone's employment on medical grounds.
Where a medical condition prevents someone from receiving JSA, they may well be eligible for ESA:
https://www.gov.uk/employment-support-allowance/overview
If you're in financial hardship, you might be eligible for a one-off grant from the Stroke Association:
http://www.stroke.org.uk/support/life-after
The Stroke Association can also offer other advice and support. Start here:
http://www.stroke.org.uk/