Because 12 days after the birth, Elizabeth was probably considered quite safe. AFAIK eclampsia or puerperal fever would strike within a week of the confinement, and in fact it is possible that Elizabeth's death was not a direct consequence of the birth.

In fact (looking back), Elizabeth's health and/or looks are mentioned in every letter of this group read so far, except the second. This seems to indicate real concern, and with reason. A pregnancy was always dangerous, especially in this case: she was having her 11th child in 15 years!