If the bits are there and still in one piece then it's fixable, but not easy. The cover piece is not attached by a hinge, but is only held in place by the spring that holds it shut. If you disassemble the lock it is possible to rebuild with spring and cover in place. But it might be the spring and/or cover is broken or lost.
An easier option might be to find a new lock barrel and re-jig the tumblers to match your key. You could probably just transfer the tumblers from your current lock
I know how annoying this is, as the lock cover on my E38 is lazy, and only parts closes. Grrr