But we really don't know how long Elizabeth stayed in bed after the birth of this child, and we know very little else about her health. She was suffering a cold when Jane visited in the Summer -- that could also be related, I suppose.

Cassandra had (evidently) sent Jane a report that Elizabeth was recovering well, about a week before Elizabeth died. Again, we don't know what that means -- but it could mean that she had already been up and about. On the day she died, Elizabeth ate and then became violently ill, whatever that means, and died within half an hour; this is according to an eye witness account. That's about all we know for sure.