From the LDS point of view, you don’t really even have to get into the discussion of Faith vs. Works. The answer is very simple. They (demons and Satan) are actually not eligible for salvations no matter what they do.
In the Pearl of Great Price you will find the Mormon account of our “pre-mortal” existence. It tells of a grand council in Heaven before the world was created. At this meeting, two plans were presented to God. The first was from Satan who had a plan that provided a way for everyone to be redeemed and not one soul would be lost. The cost of his plan would be our agency to choose for ourselves and he would take all of the glory for himself. The other plan was from Jesus who offered himself as a savior, leaving our agency intact and giving the glory to Heavenly Father.
God chose Jesus’ plan and Satan rebelled, becoming the devil and a war started. Satan managed to lead 1/3 of the host of heaven away from God and they were all cast out of heaven, becoming demons. Once they were cast out, they lost what is called their “first estate”. In the October 2006 General Conference, Elder L. Tom Perry said the following about the first estate.
“We first heard about the plan of salvation before we were born, in what the scriptures call our first estate (see Abraham 3:26). What occurred in this first estate is dimly understood, but we do know that we lived there as spirits, children of our Heavenly Father, and we made certain steps of advancement to prepare for the opportunity of housing our eternal spirits in earthly bodies.”
He goes on to say,
“We lived as spirit children of our Father in Heaven before we were born on this earth. We were not, however, like our Heavenly Father, nor could we ever become like Him and enjoy all the blessings that He enjoys without the experience of living in mortality with a physical body.”
As Elder Perry pointed out, those who have been or will be born on this earth kept their “first estate”. Once Satan and his followers were cast out of heaven, they lost their “first estate” and they were no longer able to be born into a physical body and learn to be like God. No amount of works or faith can change their eternal fate and they will never experience salvation.
If this is the Mormon answer to the question and under their own doctrine, such salvation is not possible for Satan and his demons, why then even ask it? By asking it, they unwittingly poke a hole in their own argument of why “faith alone” is not true because their alternative is not a possible solution.
Do Satan and his demons believe in God? They certainly do. They are part of the unseen spiritual realm which I think has access to both the spiritual world and ours which is something hard for us to comprehend because we only have visual access to our world (with very few exceptions). I imagine this takes them out of the faith category and puts them into the knowledge one. They KNOW who Jesus is and that is why they fight so hard against what God is doing for and through His people. Unfortunately for them, Jesus came to save the human race and they are not part of that.
I have heard it said that Satan cannot be saved because he will never submit himself to God’s will which is part of being “born again”. Knowing what I do about his character and goal, I would say that this statement makes perfect sense to me.