It might be worth having the back of the watch removed to see if there is anything engraved on the movement.
You might find that cleaning the watch gets is going, although the advice of the watch repairer should be asked for on this, as it might be very worn and in need of new parts which are unlikely to be available.
If it works and there is information on the movement, you might be able to get some information on the watch from the maker. Visit their web site (www.jaeger-lecoultre.com/) and make an inquiry, giving them all the information that you have.
Answers & Comments
It might be worth having the back of the watch removed to see if there is anything engraved on the movement.
You might find that cleaning the watch gets is going, although the advice of the watch repairer should be asked for on this, as it might be very worn and in need of new parts which are unlikely to be available.
If it works and there is information on the movement, you might be able to get some information on the watch from the maker. Visit their web site (www.jaeger-lecoultre.com/) and make an inquiry, giving them all the information that you have.
An interesting problem!
Good luck.