Replace fluorescent bathroom light with an LED (PL26E/H/841/LED)?
The spec sheet specifies Plug and play.  That means you do not replace the ballast.  The opposite is "direct wire", which is where you do. 

If it were a direct-wire unit, it would specify which pins to wire to. 

The spec sheet also explains why it failed initially, in the upper right it says "Feit Electric".  You can tell from the errors on the data sheet they're just an importer of goods they buy across the ocean. 

Seriously, though, it needs the ballast to be working.  I have a feeling the ballast was defective.  Regardless, it's finished, now that you cut the wires so close. (I have to imagine you kinda knew it was defective).  Toss it in the trash (it's electronic, no PCB).  Save the screws/nuts that hold it down though.  You may need them badly, and finding random matches at the hardware store is no fun. 

At this point you have 4 options: 


find a direct-wire LED that fits in that connector (good luck)
replace the ballast and get a plug-n-play LED or actual fluorescent tube
rework the fixture to use a different style of connector that is good for LEDs
replace the fixture 


As for the probably-fried one, take it back to HD, tell them it's defective, get your money back, and stop shopping there. This kind of wild misapplication always traces back to big-box stores. 

From now on, deal with real electrical and lighting supply houses.  They sell to contractors who want the best price on products that won't generate service calls.