Oh no.. I'm not playing this game with you.
And I'm not going to let you paint me as someone who sees demons everywhere. Especially not when I've pointed out more than once that it's not certain that it IS a demon.
How old is the woman plagued by the demon? Also, have there been any significant changes in her life prior to her troubles, good or bad? These are important, psychologically and spiritually. I am not mocking you or any other such nonsense, I assure you, I will approach this from all angles. From what you described in your first post, this sounds almost more like a haunting than a possession. That she moved to a new house is critical, to test for the difference between a haunting and a possession, if she leaves the house this crap should not happen to her until she returns. In a possession, it follows you no matter where you go. Any recent deaths, both in or around the house as well as anyone she knows would also be good to know about. The occult stuff that far back shouldn't do anything, especially considering she didn't do it in the house.
I also say haunting more than exorcism because the more attention given to it, the worse it continued to get. Hauntings are fueled and worsened by strong emotions and attention, so the more you pay attention to them, the worse they get. Change triggers them too.
As far as I am aware, you can't really exorcise hauntings, but you can live with them so long as you are mindful of it. Most will calm down to a livable degree so long as you keep the environment stable and stop paying attention to it.
But if it is really one of these two things, it is critical to find out which one because all that I said is precisely what not to do with a possession. Possessive entities seek to isolate and tire their victims, which they attach to and follow wherever they go. Houses can be possessed, as can other objects, and worse possession can transfer, unlike a haunting which is confined typically to one item or person.
As for the Jesus thing... it is actually fairly common for violent hauntings to react to that stuff similarly to how you would expect a demon to, especially if it fits the religion of the geographic area. A thing to try would be to bring in a religious article that isn't Christian. Perhaps a little Buddha figure, or a Hindu luck charm? Demon related hauntings and possessions react to those differently than hauntings sourced differently, though I won't tell you how least the crafty things try to imitate your expectations. Not until you see the results or hear about them. I warn you though, doing that could make things escalate very quickly, and you should probably have some cameras around.
In order of danger from greatest to least:
Demonic possession: worst "cases", still of course uncomfirmed as are all of these really, the possessed person dies and the demon attaches to a new victim. Will usually target one person far more heavily than others who live in the house. Exorcism is just as dangerous as the possession.
Demonic haunting: can make the environment unbearable to live in, but usually won't result in human death. Wish I could say the same for smaller animals. Impressionable people targeted. Exorcism can occur without much resistence, if done right.
Other possession: usually will be more like an attachment than anything else, what makes it bad is that the possession can move around, leaving victims without escape. If violent, will target one person heavily, or will target those who get close to the object it possesses. Exorcism has mixed results. Death exceedingly rare.
Other haunting: restricted to one general area, and exorcism is usually not warranted. Have periods of extremely heavy activity following changes in the environment, and declines when the environment remains stable. Can eventually fade away on their own.
I am going off of years ago when I hardcore actually believed in this stuff.