The location you show on your Spyder is for an OEM incandescent light. The factory design already has a 12v source to a fixture in that location.
Take heart. This should be a relatively easy job.
Did you do that splice job ?
All that tape and splicing is a very poor job. Probably wires twisted together and then wrapped in too much tape.
The twisted wires are subject to moisture and can cause too much resistance, stopping the flow of electricity.
The first thing I recommend is get access to an electrical tester.
AND a soldering iron.
Next, I'd take my LED and connect them to a 12 v source directly to make sure they still work.
You should be able to use your new electrical tester to confirm the + and - terminals of your 12v source. This way you can make sure you'r not using a dead battery.
As others have said, the LED is dependent on getting the + and - on the correct terminals. If the lights don't work, try reversing the polarity.
God knows, reversing the polarity always worked on Star Trek.
If they still don't work, and you've confirmed the 12v source is active, then the LEDs are broken and you have to get new ones.
If they do wok, I'd make sure that I clearly marked which wire I connected to the + terminal during the successful trial. I usually use a small piece of tape with enough left over to make a tab, and mark it with a + or -.
Next, I'd start working om the messed up connections at the fenders.
Remove all the spliced pieces of wire, right down to the Factory harness.
If you don't have enough space to hook the new light fixture up, you'll need to make a correct splice.
Using your electrical tester, find the + and - wires and mark them. You'll have to turn your Spyder on in order to get the voltage on the harness connections.
Turn the bike off, and then connect the lights up, making sure you have the + to the + and the - to the -.
Turn the bike back on, and the lights should come on.
Shut the bike off, and then solder the splice to make sure you maintain a good connection. You can then put the tape back on to cover the splice.
Do the same on the other side.
One point: When I replaced my LED headlights, I found that BRP had mysteriously reversed the + and the -, so you'll need to confirm the polarity first when working on the opposite side.
I know it's been long winded, but I hope this helps.