First of all, I'd suggest looking at the grub problem, before doing anything else. What does it do? does it show an error? If yes, which one?
Is it possible to reinstall kubuntu, without touching what is on the machine?
Without touching what is on the machine, no. Without touching your /home yes, but its easily done only if its on a separate partition, otherwise you can't reinstall from scratch (format root partition).
In any case, before reinstalling kubuntu, try reinstalling only GRUB (I believe there's a utility on the disk to do that)