I cannot understand the relation between your mother's ill health and divorce decree expeditiously in your hand. Is your divorce decree cure her from her ailments? Or she is interested to get you tie with some other girl soon you get a divorce decree?
If you are not in the thinking of another marriage, then do not accept your wife's condition. I am not giving this advice just keeping your or your wife's interests into consideration. My main concern is only for your child. For her healthy growth she needs mother's love as well as father's love. So convince your wife that you will give custodial rights of the child to her but she shall give visitation rights (temporary custody) of one month in summer vacation and 15 days in chirstmas vacation. If she accepts for this, then only go for MCD. Do not put at stake the interest of the child for getting quick divorce. Again, if you are in the thought of another marriage, then give your wife full custody, because your daughter may prefer to live with her biological mother 365 days of the year instead of living 45 days with a step mother.