iPhone 5 Won't Turn on Unless Plugged In After Battery Change

TLDR: iPhone 5 only half work after battery change, me sad. Check video or pics below to see what it's like to be sad boy.

iPhone 5 had a swollen battery; home button wasn't even working and screen was starting to come out on the left side. After deducing it was due to a swollen battery, I changed the batteries for a new one.

I bought the batteries from here: http://storefarm.naver.com/iphonehousing...

The reviews all claim it's a "certified" battery made by Sony. Being a sad boy, I can't tell the difference anyways.

Please check out these photos if it'll help you experts.

http://imgur.com/a/9N2Uk

The following is a summary of events and things I've tried:

-After I changed it, the home button worked again, and everything as far as I could tell worked (but I hadn't put the SIM card in yet). As it was late I went to sleep.

-~12 hours later (the morning after), I slapped the SIM card in, and it still worked fine but the 3G data wasn't displaying, so I did a hard reset.

-After the hard reset, it would turn on to the lock screen for a couple seconds, before turning off.

-IF plugged in, it will work PERFECTLY.

-If I unplug it, it would turn off in a few seconds (as if the power was cut off, not as if one regularly turned it off)

-Tried replugging everything according to the iFixit guide several times.

-Now it's pretty much the same but sometimes displays the "need to charge phone" icon. Also, it won't even go to the lock screen anymore.

Some other strange behavior:

-100% of the time when I leave it on timer while plugged in, then unplug it, it'll stay on for as long as the timer is sounding off (check end of video below to see what I mean).

-Once (after doing a hard restore from iTunes), it lasted an entire hour without being plugged in. I could play an entire round of PvZ 2, surf on Safari and everything. But the moment I try to make a phone call or receive a phone call, it turns off.

-It did it once after that too, but without the hard restore.

Check out the YouTube video here to see:

https://youtu.be/s4jFe21W5a8

EDIT: Forgot to add - Before the battery change, the reason I became aware of the swollen battery in the first place was because it would turn off randomly after 1-2 hours of usage at 70-80%. Sometimes less and only kept getting worse. By the time I got around to changing the battery it probably lasted a whole 30 minutes of usage on average before turning off.