Hi,
You'll be able to find your ilok user ID from avid - At some time they will have put the license in your ilok account so they will have a record of that.
Once you've got that you can have ilok reissue your password to gain access to the ilok account.
That said, it doesn't really help you. The license you own is on a broken ilok so you can't move it anywhere.
You'd need to contact ilok (once you have your credentials) and tell them what happened.
I don't know if they charge for the service or not, but I know they will issue the license to a new ilok if they are satisfied that the original license was legit and the broken device has been surrendered.
If you physically broke the thing, I'm guessing you cracked/snapped the USB connector off? That would mean it's probably repairable. There's only 4 pins in those things.
If that's the case and it was my ilok, I'd repair it myself, move the licence off the ilok, and buy a new key. That might be no use to you but I thought I'd mention it.