I have been telling her to pray by herself and that we cannot pray in Jama ah. You are correct. It is not permissible for a non-Mahram man and a woman to pray in Jama ah. Euthanasia is not allowed in Islam. Islam considers human life sacred.
Life is to be protected and promoted as much as possible.
Killing is allowed only in a declared just war situation when the enemy comes to attack, then killing of the enemy is allowed for self-defense. The court of law may pass a death sentence against a person as a punishment for some crimes such as premeditated murder or other serious crimes. However, there is no provision in Islam for killing a person to reduce his pain or suffering from sickness. It is the duty of the doctors, patient s relatives and the state to take care of the sick and to do their best to reduce the pain and suffering of the sick, but they are not allowed under any circumstances to kill the sick person. The sick person also should patiently endure the pain and should pray to Allah.