unable to boot: Mac HD verify error - invalid index key

Thank you in advance for your assistance. Please provide a detailed response. There is a lot I have to learn about computers.

I allowed my macbook air mid-2013 os x 10.10.2 to update yesterday. I didn't look at what the update was (just saw the notification and clicked restart). It shut down, but did not restart. When I try to boot, the loading bar slows and tops at approx 35%, and the computer shuts off. I can boot in recovery mode. In disk utility, on the left side the first line is Macintosh HD. I can verify this without errors. The second line is also Macintosh HD written in lighter grey. I cannot interact with this until I click unlock and type in my password. After doing so, it is written in dark black, and I then have the option to verify disk. Doing so, I am given the follow error in the log:

(in red) Invalid index key.

(in red) Invalid record count.

(in red) The volume Macintosh HD could not be verified completely.

(in black) File system check exit code is 8.

(in red) Error: This disk needs to be repaired. Click Repair Disk.

A pop-up message says:

Disk Utility stopped verifying "Macintosh HD"

This disk needs to be repaired.

I click an "ok" box, but the Repair Disk button is not available.

At this point, I noticed the "mount" button is available. If I click this, it says:

Mount Failed

The disk "Macintosh HD" could not be mounted.

Try running First Aid on the disk and then retry mounting.

Of course, the only option in First Aid is Verify Disk, which I have already done.

I also tried reinstalling OS X from safe mode, but the installation failed. I would ideally like to fix this without losing any data, but my computer is backed up so it won't be terrible if I have to start fresh. My SSD failed on my macbook air 2012, so I hope this isn't happening again.

Thank you.