Is it permissible for a Muslim woman to use birth control medication for a long period or permanently, due to the deterioration of her health after a premature twin birth and her care for five children?

A Muslim is not permitted to use permanent birth control methods. However, if the prevention is for a specific, temporary period due to a need, then there is no legal impediment to it. The Prophet, peace and blessings be upon him, permitted coitus interruptus ('azl). The length of the period does not matter if there is a need for it.