If the 2011 has thunderbolt, a thunderbolt2 to thunderbolt3 adapter -and- a thunderbolt3 connecting cable should do it.
The tbolt3 adapter is $50 from Apple.
It might be possible to find a slightly-used one cheaper on ebay.
For a one-time use, that's what I'd do.
Tbolt3 cables aren't cheap, either, about $25 or so.
Hmmm....
IF there's an Apple Store anywhere nearby, they MIGHT be willing to perform this connection/transfer free of charge for a new iMac owner. Of course, you'd have to "carry both iMacs in" to the place, and back out again.
Another thought:
You said that YOU can get it into target disk mode?
How, exactly?
If you can get access to the drive from another Mac, why couldn't you do the following:
1. Get an external drive large enough to hold the contents of the (non working) Mac's internal drive
2. Get CarbonCopyCloner or SuperDuper (both are FREE to download and use for 30 days, my solution costs you nothing)
3. Use either CCC or SD to clone the contents of the old iMac's internal drive to an external USB drive.
4. Now, take that external drive to the new iMac.
5. Boot for the first time and run setup assistant, and when setup assistant asks WHERE you wish to migrate from, "point it" towards the cloned backup.
6. Bring over the client's relevant data that way.
Instead of spending $75-100 for an adapter and cable that the user may be used only once, have him spend it on a decent USB3 external drive, do the "cloning process" above, and then show him how to KEEP USING the old drive to back up his new iMac. This will be money "much better spent" for the future...