That is what I would do. Use either a dremel-type drill with a very fine dental burr cutter, failing that, a scalpel blade with a fine taper. Obviously, care to avoid the remaining wire.
You will probably find at least one of the conductors has blackened wires, if so, scrape them gently to get back to shiny copper, otherwise the solder won't wet it. get solder on each cut wire and then do the same on the ends of the detached plug. then solder the two together. It almost certainly won't matter if you swap the wires over in the process. Then use rapid araldite. experiment first with the colour. Kids powder paint or any fine dust the right colour will do. The araldite will have to go a little way up the insulation beyond the solder, so it may pay to 'fold' the joins back in to the cavity in the dipstick so you end up with something that looks original. If you are near Bristol, i would willingly help (as I would with any local problem.....).
By the way forum admin. In line with my views on them voting in that t**t, can you get rid of the yank spellcheck. There is and always will be a 'u' in colour.....