Unfortunately, the conduit for light niches is often badly installed, in a variety of ways. If the cord won't move, you can try forcing some wire lubricant down the conduit. But this won't help if they've crimped the conduit.
Or you can use a waterproof splice kit to connect a new lamp to an old cord. It's not standard, but you may want to coat your wire splice connector with electrical silicone, BEFORE sliding the heat shrink tubing over the top. I assume you have a GFCI on the lamp, and it doesn't take much current leakage to trip the GFCI. Using the silicone can help prevent that.
It may sound like overkill, but it might be a good idea to buy TWO splice kits, so you can practice with one, before applying the other. Note that most silicone will corrode wiring.
Also, a quart is ~57 cubic inches. If your conduit is 3/4 ID, you will need about 2 cubic inches per foot of conduit. So, a quart should do at least 25' of conduit.
Good luck!
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Gardner Bender HST-1300 Underground Cable Splice Kit for waterproof splicing of #14 - #8 AWG UF Copper Wire
Permatex 80050 Clear RTV Silicone Adhesive Sealant, 3 oz.
Gardner Bender 79-006N 1 Quart Squeeze Bottle Wire-Aide Wire Pulling Lubricant
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