[Q] How to get data off phone in recovery mode

I tried to install the 4.4 Android update on my Nexus 4 but upon reboot my phone started at the "four circles" screen. I thought it was installing correctly so I let my phone sit while plugged in, but when I checked back in four hours it was still stuck. Every time I reboot it gets stuck at the same screen.

I can access Recovery Mode but I don't have any backups for the system before I tried to install this update (really bad planning on my part, I know). After searching on Google it appears the only solution is a factory reset and installing files found on this website, but before I do that I want to pull my data off my phone. When I connect my phone to my computer via USB it doesn't show up at all under Computer.

I have the following programs installed that I think are relevant:
Team Win Recovery Project
SuperSU
Titanium Backup (with paid key)

I've tried scouring Google for days but I must not be using the right key words because I can't find anything relevant to my specific issue. I've found mention of "ADB Sideload" but I don't understand how that factors in-- I haven't found anything in simple enough terms for me to understand (I'm extremely un-knowledgable of phone-related tech talk).

Please help! Thank you!

jd1639 said:

Google adb backup xda

Sent from my Nexus 5 using XDA Free mobile app

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From what I'm reading after Googling "adb backup" you need your phone to be able to start up. My phone can't start up, it can only go in to recovery mode. Also, when I backed up my files with TWRP and restored it, it also restored the faulty installation of Android 4.4. As far as I'm reading on adb backup it will backup everything the same way, so it will also reinstall the faulty installation of Android 4.4, in my understanding.

Is that right? Or is there something I'm missing where the adb backup file can be modified while on a computer where the old system files (for the phone) can be removed so it only restores the data? Also, I've read that the adb backup does not save any SMS messages, will it still save the SMS backup files I have through Titanium Backup?