My friend Chester, master guitar builder, uses CA cyanoacrylates on all small
gluing that doesn’t call for hide glue.
 I fixed a acrylic lid for a rice cooker a while ago. Didn’t think it would take
the steam but its fine, and happy in the dishwasher too.
All the suggestions are good, but perhaps over kill. Just Crazy Glue with maybe
a tiny thin cloth patch across the crack.

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