Replace, don't splice.
Don’t bother repairing a standard extension cord. Cut it off and buy a new one.
The problem is that a spliced cord introduces a weak point. Even with a proper inline splice and heat shrink, you’re creating a spot where resistance can build up, heat can develop, and fires start. Unless you’re willing to test the splice with a multimeter and redo it if it’s off, you’re guessing.
For heavy-duty outdoor cords (10-gauge or thicker, 50+ feet), a professional repair can be worth it if you know what you’re doing. But for the $8 orange indoor cord from the hardware store, just toss it.
A $10 cord isn’t worth a house fire.