iPhone 5s Infinite BSOD boot loop [broken home button]

So I've had my iPhone jailbroken on iOS 8.1 I believe for about a year now. Everything's been great. Or at least it was. In the week leading up to the boot loop I had a few minor issues like the screen touching itself here and there, my home button becoming less and less responsive until it stopped working entirely, loss of 3G carrier settings. But all in all nothing major that rendered my phone unusable.

I went onto a website using the phone browser that promised to help me install the correct APN, and it appeared to have done so. I went to the APN settings to get a closer look at the details and that's when my phone froze. I immediately turned it off, and ever since it's been stuck in this Apple Logo boot loop that culminates in a blue flash on the screen. And it repeats until the battery dies. I can't put it in recovery mode because the home button is broken. iTunes won't detect it at all.

I've taken it to two different repair shops. The first guy said it's something to do with misplaced screws in the motherboard. Which is weird because I've never opened the phone or had any repairs done. I figured he was lazy, Googled the problem and dismissed my case without trying to help. The second guy I asked to just fix the home button, so I could put it in recovery. When I went back to him he said he couldn't get it working (recovery mode) even with the new home button, but I'm not sure if I believe that. Though he didn't charge.

Basically I'm &&^&@@. I've lost my baby. I would do anything to get it working again. Should I just replace the home button myself and try to put it in recovery? If that works will I lose my jailbreak? What the !&&* could have caused this? I haven't dropped my phone or tampered with the inner workings.

One thing that is weird though: I have f.lux installed on the phone. Again it's totally unresponsive to any button presses etc. but when it first boots up, I can see the Apple logo change colour almost as if f.lux is still working somehow. Probably not, but I thought I'd include it here just in case.

TL;DR: BSOD boot loop. Never had screen replacement or any repairs done. Home button broken. Can't enter recovery mode.