Need a correction, and I'll re-state the steps above clearly (without any of my typing "snafus"!).
First of all, it's great you have a late 2011 13" Macbook Pro, as that has a Thunderbolt port also.
Secondly, here are the steps again, with corrections:
1. With both machines off, connect the Thunderbolt cable to the respective Thunderbolt ports.
2. Boot the 13" MacBook Pro, and wait until it completes the boot process, that is, gets to the Finder/desktop.
3. Start up the problematic iMac, holding down the
T key only as soon as you press the power button. Continue to hold down the
T key. Wait until that completes.
4. You will know when step 3 completes, as the internal hard drive for the problematic iMac will show up on the 13" MacBook Pro's desktop. From there, you can use whatever disk maintenance/repair/software "repair", etc. tools you have on the MacBook Pro.
If you only have Disk Utility as the disk maintenance/repair program, that should be enough. What you'll need to do with Disk Utility is:
1. Verify and Repair the iMac's drive at the Volume level (the "top" level).
2. Verify and Repair Permissions, and Verify and Repair the disk at the partition level. Most likely, you have only one visible partition on that drive, and it would be the one containing the OS, all your software, data files, etc.
One thing I discovered the other day (via a very helpful post from another fantastic user here), the Repair Permissions via Disk Utility
only does that for Apple products. So, if I were you, after repairing the drive, I'd download the excellent freeware product called
Oynx. Besides repair permissions for
any software, it also has other useful cleaning features, along with some limited disk maintenance/repair features. You can get
Onyx from here:
http://www.macupdate.com/app/mac/11582/onyx
Careful, though, as you'll need to get the right version, which depends on which OS you are using (explained below). (Like Safari that I mention below, there are "Related Links" on that page, which can get you the correct version).
If the OS on the iMac is Snow Leopard, and you are fine with using that (there have been 4 subsequent releases by Apple of the OS, the latest being Yosemite (in fact, an update to Yosemite, 10.10.3, came out 2 days ago)), make sure it is version 10.6.8. If it's anything less (like 10.6, for example), you can upgrade that (after all the repairs are done, and the machine is running fine) to 10.6.8 by downloading it from here:
https://support.apple.com/kb/DL1400?locale=en_US
Along with that, and assuming you use Safari as your browser, you'll need to get the latest version for Snow Leopard. You can get it from here:
http://www.macupdate.com/app/mac/15675/apple-safari
You'll need Version 5.1.10, which is listed under "Related Links".
If you have a different OS on the iMac, let us know, as it might not be the latest version.
If you run into problems trying to repair the drive, not sure how informative the messages are from Disk Utility regarding that. If that happens to be the case, it would be better if you had a more robust disk maintenance/repair program like Disk Warrior or TechTool Pro, even for the future. I have both products, but I use TechTool Pro just about exclusively as part of my disk maintenance/repair/backup weekly processing. It does help with me keeping both of my machines "mean and clean".
Finally, assuming you are not doing any backups, I would
strongly suggest you start doing that for
both of your machines. With Snow Leopard, there is no built in Apple program to do that. Later versions of the OS include
Time Machine, which is "OK". But, that software does not create a bootable system (and thus an easy way to recover from a disaster). The two best products for making a bootable backup are either SuperDuper! or Carbon Copy Cloner. Both work very well. You can use SuperDuper! for free, although it will run somewhat slower than the paid version, and it does not have some additional (and useful) features like the paid version does. For the paid version, SuperDuper! costs $27.95 US, whereas Carbon Copy Cloner costs $39.99 US. You can get the "free" version of SuperDuper! from here:
http://www.macupdate.com/app/mac/13803/superduper
I can tell you from direct experience that SuperDuper! saved my "hide"
twice when I had a couple of issues with my late 2012 Mac Mini. And, given that you have an external hard drive, you are actually just about ready to do the backups (you'll need to partition that external drive with at least two partitions, so that you can use each of those for backing up each of your machines).
For future reference, you should develop a process to performing "periodic" disk maintenance/repairs/backups. There are various tools you can use for that, and no one method is "better" than another. I use one method every week for both of my machines, and it involves 4 steps (for the 2nd through 4th steps, I use Onyx, TechTool Pro, and SuperDuper!. I am also constantly removing stuff from each of my machines that I no longer need (especially EMails)). But, some other folks here use s different approach (and some different products, with success).
There is also some other useful (and free) software that is available to maintain the health of your machines, but let's see where all this gets you.
Finally, it would be good to know the version of the OS on the 13" MacBook Pro. Also, once you have OS 10.6.8 on your iMac, you can upgrade to any of the 4 newer versions of the OS for free. In fact, you can upgrade all the way to Yosemite, Version 10.10.3.
Good luck, and let us know how it goes.