Hello all. Just did this last week to my mid-2010 iMac11,3 while replacing the hard drive with an SSD. Very annoying. Fortunately, I've been in the electronics repair business for many years, so I wasn't terribly troubled... just didn't know for sure that I could find the necessary replacement part. As it happens, a gentleman in the UK seems to have plenty for sale on eBay, or you can buy them directly from Digikey Electronics here in the states. From either party, they're in the $18 to $20 price range, and that includes shipping. I ordered several, just to have them on hand. I got mine several days after ordering and had the Mac up and running in a couple of hours.
With that being said, I'm aware that another member of the forum in PA has graciously offered to replace these parts professionally for a modest fee, and if he's still interested in doing that, I certainly recommend you contact him as to his current availability. In the event that you're not successful in that regard, I would encourage anyone who's interested to give me a shout and I'll be happy to replace that connector for you. My price would be a bit more than he had offered several years ago... I've got to make a living, too... but I am a professional and I'd be happy to do it for about half of what I've seen quoted from other outfits mentioned in this thread. Shipping, of course, would be at your expense. Or, if you live within a reasonable driving distance of Fairfax, VA, you're free to drop it off and go shopping for a while, then come back and pick it up an hour or two later. We'd need to co-ordinate in advance, of course.
A word of warning: Do NOT attempt this yourself unless you're reasonably skilled in this sort of thing. There's an excellent chance that you'll make things worse and end up needing to replace the entire board, a much more expensive proposition. No offense intended, but I've seen some photos in this thread of attempted repairs that make me cringe. You wouldn't attempt your own open-heart surgery, would you? Let the folks who have the skill set handle this one, or be prepared to do further damage; the choice is yours. And if you've lifted a trace or two, that takes the repair up a notch, and may well make it impossible to do.
Should you care to take advantage of my offer, please do NOT post here. I won't be browsing this thread on a regular basis, as I'm busy with grandchildren and life in general. Send me an email to BOB.HAM (at) GMAIL.COM with the subject line "Broken Display Connector iMac 27 Inch" so as to get my attention, and I'll be happy to write you back with further info.
Best wishes and good luck in your endeavors!
P.S.- There may well be other iMacs that use this same connector; I've not checked other models, but I believe I've heard reference to that. If you find that to be the case, I can handle those, as well.