A question before going further:
Why not just put an SSD into the 2011 MBP and "keep on going"?
This is a 15-minute job with the right tools and ANYONE can do it.
Having said that:
What you could try to do:
1. Boot and work from the wife's MBP (the only working computer you have).
2. Get a USB flashdrive -- 8gb should do, 16gb is better
3. Download the El Cap OS installer from here:
https://support.apple.com/en-us/HT206886
https://itunes.apple.com/app/os-x-el-capitan/id1147835434?mt=12
4. Create a bootable USB installer using DiskMaker X, which is available here:
https://diskmakerx.com
(it's free)
5. Boot the wife's MBP with the USB installer
6. Before you install the OS, use Disk Utility to ERASE the ENTIRE drive to Mac OS extended with journaling enabled
7. Then install the OS. It may involve one or more reboots and will take some time
8. When the install is done, you'll see the setup screen "select your language"
9.
Connect your TM backup before going further
10. Go on with the setup. At the appropriate moment, setup assistant will ask if you wish to migrate data. YES, you want to do this
11. Select the TM backup, and wait for setup assistant to "digest" everything. BE PATIENT
12. I suggest you migrate everything.
13. Now let setup assistant do its thing. Again, be patient.
14. When done, you should see your login screen, looking as it did on the 2011 MBP.