Whilst experimenting with another project I ended up cutting a length of LED tape between the intended cutting lines to fit it within the project in question (don't fret, I'll reveal it when it's done). The piece didn't work after this cut as it severed multiple tracks which I didn't know about, so I was left with a strip of LEDs that wouldn't light. I hung on to them anyway and then realized that they could be reused in another project so I decided to document the relatively easy process of rewiring single chips from LED tape for singular use.

Supplies

LED tape, I used warm white 12v tape. The chips might vary between brands and tape types so you will want to do some testing before buggering up the chips.

Wire, smaller gauge and any colour you please.

Solder.

3v power supply, I used the venerable CR2032.

Soldering iron.

Scissors, I used Magnusson Electricians scissors which have an inbuilt wire stripper.

Step 1: Cut an LED Chip Out.

Remove any waterproofing if necessary. Locate the chip you want to extract and, using a pair of scissors, cut the tape on both sides as close to the chip as you want. You should be left with the single chip which is still soldered onto the tape. The adhesive backing can be used but in my case it had perished and wouldn't adhere to anything.

Step 2: Do a Little Probing Around.

If you can, finding a datasheet for the chip will pay dividends, I decided to figure it out myself. Using the 3v power supply (start off at 3v to limit the damage should you mess up) probe the connections of the chip to find the pin-out. In my case all of the cathodes and anodes are separate lined up each sides. Thus, a long wire soldered across each side will power all three LEDs in the chip (you can wire up each LED separately). Chip pin-out will vary.

The photo shows a cutout in the lower left corner of the chip, the pins located on the side of the cutout are the individual cathodes.

Step 3: Attach Leads to the Chip.

If you are planning to control each LED on the chip separately (RGB LEDs are a notable example) then this will take a little longer. I wanted to connect all the cathodes and anodes together, strip a piece of wire and give it a good tinning, then solder it onto the pin(s) you need.
Repeat this for the required amount of connections. You might need to tin the chip leads a little, I didn't but it all depends on the quality of the solder that the tape was assembled with.

Step 4: Test the Chip.

Using the suitable power supply you used earlier, power up the chip from its new leads, if the chip lights up in the way you intend then all is good. If not then check the polarity before you start pulling the leads off. If an LED is out, most likely a loose connection is to blame. If you've wired up the LEDs separately, check for solder bridges and unintended joints.

That's it! I know, a little short. We've all got to start from somewhere. Keep an eye out for more bits and bobs.