Answer:
See the image for marked congruences.
1. JM ≅ LM | Given
2. △JML is isosceles | definition of isosceles
3. ∠MJL ≅ ∠MLJ | isosceles triangle theorem
4. m∠MJL = m∠MLJ | definition of ≅
5. JK ≅ LK | Given
6. △JKL is isosceles | definition of isosceles
7. m∠KJL = m∠KLJ | isosceles triangle theorem
8. m∠MJL + m∠KJM = m∠KJL | adjacent angle theorem
9. m∠MLJ + m∠KMJ = m∠KLJ | adjacent angle theorem
10. m∠MJL + m∠KJM = m∠MLJ + m∠KLM | transitive property of =
11. m∠MJL + m∠KJM = m∠MJL + m∠KLM | substitution
12. m∠KJM = m∠KMJ | subtraction
13. ∠KJM ≅ ∠KLM | definition of ≅
14. △KJM ≅ △KLM | SAS theorem
15. ∠JKM ≅ ∠LKM | CPCTC
16. KM bisects ∠JKL | definition of bisector