you can use the console to do it.
open the console, click the door, so on top of the screen it says "Door" and some number.
then simply type "unlock" into the console.
as you close the console, the door should be unlocked, even if you broke the lock in the first place.
same thing works for computers you tried to hack and you got locked out of.
in short: "unlock" works for EVERYTHING that is locked, either computers or doors of any kind.
2 things i'd like to add.
1) after you ~/open console CLICK on object so you get the DOOR [item id]
2) be sure youre in 1st person view tho. if you are in 3rd person view you may get a message like > "" [item id]
i had some trouble getting it to work, and someone pointed that out to me.