Monthly payment against total cost
The trade you are making
A 15-year loan clears the balance twice as fast and costs far less interest in total. A 30-year loan spreads the same debt over twice as many months, so each payment is easier, but interest has twice as long to pile up.
Why there is no break-even month
With a refinance you pay a fee up front, then wait for the smaller payments to earn it back. A shorter term is not like that. Almost all of the extra you pay each month goes straight onto the balance and becomes the part of the home you own. So compare the payment, the rate, the fees and the total interest, not a break-even date that does not exist here.
Use the real rate for each
Lenders often price the two terms differently. Enter the rate you were actually quoted for each, not the same number twice. Fees and points matter too, and they are not in this comparison.
A smaller payment is worth something
Room in your budget has real value: for emergencies, for retirement, for whatever comes up. Taking 30 years and paying extra when you can keeps that room. Just remember you may not get the lower rate a true 15-year loan would have given you.