Should the mother shorten her prayers throughout her treatment period in America, and is it permissible for her to perform tayammum due to her inability to perform wudu, and what is the method of tayammum?

If the mother intends to stay four days or more in the place where she has arrived, she is not allowed to shorten the prayer (qasr) and must pray in full (itmaam).

Regarding ablution (wudu), if she is able to perform it, she must do so, even with assistance. If she is unable, she may perform tayammum (dry ablution) and pray according to her condition, for Allah Almighty says: {Allah does not burden a soul beyond its scope} (2:286) and {So fear Allah as much as you are able} (64:16).

Ibn Taymiyyah said: "The sick person prays according to his condition... what he is unable to do is waived for him... and he does what he is able to. If he is able to purify himself with water, he purifies himself. If he is unable... he performs tayammum and prays, and there is no need for him to repeat the prayer."

It is permissible for a sick person to combine prayers, either by advancing or delaying them, even if he is a resident, should he need to do so.