If in Illinois, the Mormons become a very wealthy demographic. In the 1840s, prior to the Mormons being driven out, Nauvoo was about the size of Chicago. So, let's assume the Mormons do not get driven out.
For them not to be driven out, you probably need to either avoid the tensions, or have what happened in Missouri shock the country's conscience (maybe a worse
Massacre at Haun's Mill). Historically, 21 people were killed. Make that death toll in the triple digits - or even wipe that settlement out. Once the country's conscience is so shocked, it is likely that Joseph Smith survives 10-15 years longer. He might still be martyred, probably because he held
abolitionist beliefs, but by then, who knows what - or who - he may influence.
That makes it a major power in the state. So much so, that I think they end up creating a "Mormon Corridor" through most of Southern Illinois, Iowa, northern Missouri, and they may start to even go further west or north. In fact, a strong Nauvoo opens an interesting possibility. Perhaps Abraham Lincoln becomes the first Mormon president?