All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, we ask Allaah, The Omnipotent, to cure your relative from all his illnesses as Allaah is over all things competent.
As regards seeking treatment in this hospital, in principle this is permissible because it is permissible to seek treatment from the non-Muslims or in a hospital run by non-Muslims. However, if there is a Muslim hospital where the treatment is free and there are specialist doctors who can provide the needed treatment, then it is more appropriate to go there. That is because being treated in the other (non-Muslim) hospital may include some prohibitions, as the people who run it may have the purpose of spreading some ideas that are contrary to Islam to influence the belief of the Muslim among other missionary purposes.
Allaah Knows best.