When I meant 'continuity' I meant checking for closed circuit. The multimeter has a diode continuity test that makes a buzz noise when the probes are connected electrically (such as touching both probes to a metal bar). I didn't check for resistance as I figured whatever figure the meter would provide wouldn't provide any insight to what the problem was (as your post explains), but you answered my question anyway (about the open circuit.).
So, do you think I should put a whole new electrical system in, or just replace the cap and see if that makes a difference first?
I think I should also mention that the spark on the kill lead does not look very potent (or enough to be HT) as it is a tiny light orange spark rather than white/blue. I get a similar spark with other trimmer motors that are working fine, when the kill switch is broken and the lead must be earthed to stop the engine).