If you use Google and select images and enter GU10 Base you can see pictures of the base fittings with wires - you can buy a new ceramic base (Amazon £0.70) with wires and fit it yourself - to replace the one that seems to be damaged.
You might turn the power off and try to manipulate the broken part away from the base - having seen on Google how they fit together.
The lights will be wired in series - there is no chance of a fire - if the broken socket presents any sort of problem the fuse will pop as a safety precaution - fire is an impossible outcome.
Peccavi
July 2013