I have a giant peace lily, the kind with leaves like 8 x 16 inches…anyway, one of the leaves has a broken center stem with a gash in it, and some of the other leaves have gashes, but are otherwise green and healthy. Is there a way I can “sew” them back together or get the plant to repair itself?
No.
What he said.
If thy leaf offends you, cut it off.
Pffft, these people ^ have no creativity. Haven’t you ever heard of stem cells? Plants have stems, so get a syringe, extract some cells from the stem and inject them into the leaves. QED. 
Thats too bad. I know you can graft some plants together…how come the leaves wont “graft” back together?
To elaborate: you might be able to stitch the leaves back together, but there’s really no need to, unless you want to experiment. I’ve seen plants heal themselves in nice ways. But, it’s way more progressive to just cut off the damaged leaves, and the plant will respond with new shiny leaves Pronto. Especially a Peace Lily, a time-honored houseplant, which means it can take a heaping helping of human neglect.
Cut the leaves right down to the beginning of the stem, and watch the new leaves replace them quickly. Plants are not static— they want to survive, and will send up new leaves when old ones are cut.
It’s a function of where the meristematic cells in the plant are located – the roots, growing tips, etc.
Hmm. What if you used rooting hormone?