can you even jailbreak with broken optical drive ? I know for a fact it doesn't work without :p
what model is the broken one? (40,60,80,160?)
do you have access to a second ps3 ?
what model would that be ?
check to see if the bluray drives are the same & swap them (also the board attached to the drive) just until you get the patch installed (you don't have to sign in to psn, just start mw2 with internet connection)
if you cannot use a 2nd ps3, what region are you in ?
find someone on here that has your region & JB, get them to upload the folder

but i'd try the bluray drive first