Culzean rightly points out that the Jazz's electrics are very well protected BUT reading the symptoms sounds like the old classic of one cylinder not firing. The very low revs and then picking up. Does the engine sound rough when first starting and then settle down as the revs pick up? May be a shot in the dark but by bitter experience HT parts can be faulty with intermittent working. First start takes a lot.
Yes I remember the days when given a mist you could see a fantastic electrical display over the cars or motor bikes HT system. Holts used to sell a spray to waterproof your electrics.
Off at a tangent. 1963, many of you too young to know but that was a bad bad winter. I was 'courting' My now Wife lived 25 miles from my home. One night, leaving my Fiance, got my M/C to start eventually, gearbox was a pig I honestly think the oil was nearly frozen solid, good old EP80 and when I got home I was frozen to the blessed bike. My Mother next day asked what was the crash last night when you came home. Reply me, I was frozen to the bike and I couldn't get off just fell over. I hope I am never as cold as I was that night again.