Does Jesus always heal Christians from any sickness?
There is no record in the Bible of anyone who came to Jesus to be healed and was not. It is possible that he healed everyone who came to him asking for healing.
Now that Jesus is not physically present with us we have no way of knowing if a person is healed as a direct and miraculous result of seeking healing from Jesus, or whether it was going to happen anyway, or whether God worked in a non-miraculous way to intervene in the healing. We do know that people, even Christians, die. Therefore they are not always healed from any sickness. It may be that God chooses to relieve them of pain and suffering by allowing them to die instead of healing them.
There is one example in the Bible of a Christian, the apostle Paul, who was not healed of a sickness. See 2 Corinthians 12:7-10. God chose not to heal him, knowing that it would make him a stronger Christian if he did not.