There is nothing wrong, as far as Islam is concerned, to pay a portion of the zakah money to pay the rent of the mosque to serve the Muslim community. According to the case in point, taking a portion of the zakah money is necessary as it is clear that the donations collected are not enough.
Sheikh Ja`far At-Talhawi, a prominent Al-Azhar scholar, states the following: First, it is permissible that you dedicate part of the money of zakah to cover the costs required for renting that mosque and preparing it for practicing the Islamic rituals. This actually falls within the category known as fi sabil illah or “in the cause of Allah” mentioned in the following Qur’anic verse: (The alms are only for the poor and the needy, and those who collect them, and those whose hearts are to be reconciled, and to free the captives and the debtors, and for the cause of Allah, and (for) the wayfarers; a duty imposed by Allah. Allah is knower, Wise.) (At-Tawbah 9: 60)

Second,`imarah or maintaining such places (of worship) whether through building or hiring them is a sign of iman (faith). Allah Almighty says: (The mosques of Allah shall be visited and maintained by such as believe in Allah and the Last Day, establish regular prayers, and practise regular charity, and fear none (at all) except Allah. It is they who are expected to be on true guidance.) (At-Tawbah 9: 18)

Third, maintaining the mosques of Allah (`imarah) denotes two things:

A. The acts involved in establishing and repairing a place for worship, and this includes hiring a place to use as a mosque.

B. Regularly attending and visiting places of worship to observe prayer therein, as the Prophet (peace and blessings be upon him) said: “When you see a man regularly attends to mosques, confess to his iman (faith).” (Reported by Imam Ahmad in his Musnad, At-Tirmidhi, Ibn Majah, Ibn Khuzaymah, Ibn Hibban and Al-Hakim).