Broken lines can be repaired by splicing with a section of braided mono of sufficient diameter to insert the fly line. About 8 inches of braided mono, usually 35lb or 50lb usually works. Insert ends of fly line into braid so they meet in the middle. Secure each end of braid with a nail knot made from 10 to 12lb test mono or from a spectra (gel spun) braid like spider wire, power pro, etc. Coat nail knot with pliobond, loon knot sense, or similar flexible coating. If a floating line, a tiny tab of super glue applied to the broken or cut ends of the fly line will help prevent water from wicking into the core of the line.

I have repaired many lines like this and they work fine though there is a bit of noise when the repaired section goes through the guides.