However, in the second example, it wouldn't make any sense to be admitted to hospital "having suffered from" anything. That suggests that you had suffered from the problem in the past but not now. It would be a bit late to admit you to hospital for something you suffered from in the past. She was admitted to hospital at the same time as suffering inflammation of the ovaries. That is why she was admitted. She was in the process of suffering the medical problem and was admitted to hospital as a direct result.
To use "having suffered" in that context, you could say something like "Having suffered from inflammation of the ovaries for several years, it was incredible that she was still able to compete at an Olympic level".
Having taken off his hat, he entered the room = He took off his hat (somewhere else), then he entered the room.
She drove to the church, having dropped her children off at nursery = She dropped her children off at nursery, then she drove to the church.
Taking off his hat, he entered the room = He entered the room while taking his hat off
"Having + past simple" = an action already done, and then followed by another action
"Verb + ing" = two actions happening at the same time.
Student or Learner