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The corrosion leakage, the brownish stuff that comes out of old batteries, has to be cleaned out quite thoroughly. If the battery compartment terminals still have any trace of brownish or less-than-silvery or less-than-coppery, on the parts that contact the battery's ends, then you need to clean them even further.
A light application of fine-grit sandpaper to the ends of the compartment terminals might even be justified... whether that's needed or not, you should in every case, clean the terminals with cotton swabs or cotton waste moistened with alcohol, rubbing alcohol, or even cheap whiskey... until they are shiny, period.
(Then let them dry 5 minutes + , before putting batteries back in, with correct orientation RE + vs. - poles in the compartment, and then trying again.)
If corrosion got deep within the latch/lock/etc., beyond the battery compartment, then you may have other problems too. But try the above, very thorough, cleaning of the battery compartment terminals, then see if that's enough to make the keypad light up again.
Let me know how it goes, and what you think - I will offer further advice too.