jmc

Since $JM$ bisects $\angle J$, we know that the measure of $\angle KJM$ is $60/2=30$ degrees. Similarly, since $MK$ bisects $\angle K$, we know that the measure of $\angle JKM$ is $30/2=15$ degrees. Finally, since the sum of the measures of the angles of a triangle always equals $180$ degrees, we know that the sum of the measures of $\angle JKM$, $\angle KJM$, and $\angle JMK$ equals $180$ degrees. Thus, the measure of $\angle JMK=180-30-15=\boxed{135}$ degrees.