How do you repair a cracked plastic seat on a pool chair?

They have patches for inner tubes wherever they sell bicylcle supplies in stores. These work well.

lots of duct tape on either side of the broken part

There is a special puddy you can buy from your local pool place to fix a cracked skimmer. It dries quickly & its easy to use. You may have to do this yearly. We have had to twice now.

Repairing it is a bad idea. Many have tried, few, if any, have succeeded. The best option is to replace it with the new, improved manifold design that has an aluminum coolant crossover. The new...

Yes. I take it this in a inground vinyl pool.. With the pool drained most anyone good with fiberglass repaires can refinish..   Order new steps from your local pool supplier or from the web....