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.
Dear Sister, it is not permissible for you to work in the mentioned profession because it involves some prohibited matters, among which are touching non-Mahram men, looking at their ‘Awrah (parts of the body that must be covered in Islam) and even touching it sometimes when washing them and all this is not permissible. In principle, it is forbidden to look at the ‘Awrah of other people and it is only the spouses who may look at the ‘Awrah of each other as the Prophet said: ''Protect your ‘Awrah except from your wife or what your right hand possesses.” [Abu Daawood]
The only exception is in case of necessity, like when a doctor treats a patient and the like, and in this case, men should be treated by men, and women should be treated by women.
Also, in case there is a need that makes some people unable to wash their body, like the sick people and the elderly, then in this case also men should wash men and women should wash women, exactly like in regard to doctors treating patients.
Moreover, there is no difference in this regard between a man who is afflicted with dementia and any other man, as a woman is not permitted to look at his ‘Awrah or touch it. Likewise, a Muslim woman is Islamically ordered to wear Hijab in the presence of non-Mahram men with the exception of those whom Allaah has excluded. Indeed, Allaah has excluded the male attendants who have no physical desire in women, i.e. they are devoid of sexual feeling and not concerned about the matter of women.
However, those who have a tendency toward women and show that they feel sexual desire, then they should not be allowed to enter upon women and the women are obliged to wear Hijab in front of them. For more benefit, please refer to Fatwa 83033.
In addition to this, some scholars forbade a Muslim to work for a non-Muslim in such kind of job because it might be a humiliation for the Muslim. For more benefit, please refer to Fatwa 84798.
Therefore, based on the above, the work which you mentioned in the question involves many Sharee'ah prohibitions; so it is not permissible for you to do such work. Hence, you should look for another job that does not involve any prohibition.
Nonetheless, you get the reward for your intention to work if this is with the purpose of abstaining from what is forbidden and from asking people for money. If the work itself includes some virtues and its doer intends to get closer to Allaah by doing it, then he/she also gets the reward. This applies if a woman works in serving the elderly Muslim women and her intention is to get closer to Allaah while she gets a worldly benefit for it [salary], then she gets all of this.
Allaah Knows best.