I strongly believe that Henry Clay would have ended up as president. He was the leader of the Whigs. The Whigs where in a good position in 1841 due to the economical troubles during the Van Buren Adminsistration. So in the aftermath of Harrison's death, Clay would easily win the election while the Texas issue is no topic.
Do you think Tyler would have tried to win a term in his own right?
I think he'd try. IOTL he was adamant about the Vice President taking the full responsibility of the presidency to ensure continuity of government - and it seemed like despite all the stress it caused him, he did enjoy being president
