OK, so the problem was that the rain didn't actually drip into the gutter - oh no - it was running round the edge of the roof-felt and back along the underside of the roof, through the gap between the gutter and the wall..
and then down the wall.
..making the wall very wet.
Get a man in?
I bodged it...
And this evening I have fixed my computer by taking it apart and hoovering out all the dust.
Is there anything I can't bodge?
I think not.