WI: Andrew Jackson Dies in Battle of New Orleans

Let's say Jackson dies at the Battle of New Orleans but still is able to win the battle. Who is likely to replace him as president? How would this affect politics in America?
 
Tippecanoe seems most likely to be the first "frontier populist" president. Butterflies might well mean he serves a full term or two by not giving himself pneumonia, which would be interesting. However, policy-wise, I imagine he'd be much different to Jackson.
 
Personally, I think that Henry Clay could potentially have been ended up being president during the time that Jackson was historically. Of course, his administration would have been very different from Jackson's.
 
I think there might have been more chance of the Whigs coming around to Jackson's participatory ideas on democracy earlier on. The population pressures of the era meant that someone was going to have to champion the growing South and West.
 
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