I will supply the proof; you figure what is wrong with the proof given in the problem.
Draw a figure!!!!
Construct the line segments LK and LJ.
In triangles LMJ and LMK:
LM is common
JM = LM because LM is the bisector of JK (you might use congruent instead of equal)
The 2 triangles are right triangles because LM is the perpendicular bisector of JK
The 2 triangles are congruent by SAS
LJ=LK because they are corresponding parts of congruent triangles (again you might use the word congruent)