Hey, I'm not sure if my method can help, since I've only done it on an iPod Touch 4G on IOS6, but it's worth a try. This has allowed me to SSH into my iPod Touch and pull out any files I needed, even app/game saves, when I was in a Recovery Loop and couldn't use my iPod. After the repair, this allowed Tiny Umbrella to kick the iPod out of the Loop for me.
Since your Home button is broken, try use TinyUmbrella to see if that enters it into DFU mode if at any point in the instructions it asks you.
By the way, this is instructions for Windows, not sure how to do it on a Mac, but I'm sure you can get similar software, or maybe there's Mac versions available.
Download Putty:
http://www.chiark.greenend.org.uk/~sgtatham/putty/download.html
Download WinSCP:
http://winscp.net/eng/download.php
Download SSH_RD_Rev03B.jar:
http://www.sinfuliphone.com/showthread.php?p=781514
In that link it mentions about needing to install 'JRE-7-Windows-i586-iftw.exe' for it to work, but you can try without it to see if it works.
Follow the instructions here to get access into your iPod:
https://github.com/msftguy/ssh-rd/wiki/Windows-How-To
If /mnt2 doesn’t show, restart the instructions for Putty, and after typing in the root and password, (before typing ‘mount.sh’), enter in:
fsck_hfs -r /dev/disk0s1s2
It will go through a repair, wait till it’s done, and mnt2 should work. At this point try downloading and using TinyUmbrella to hopefully kick it out of the Recovery Loop.
Hope this works out for you.