"Lie in bed" is generally correct: It means to lie under the covers, between the top sheet and the bottom sheet. It is generally about the usual bed one sleeps in.
"¢ All she does is lie in bed all day. Isn't she bored?
"¢ You have to lie in bed, flat on your back, for ten days, or until your leg heals.
It's possible to "lie on the bed," too. This means that you are not in bed for the purpose of sleeping, but perhaps sitting on it to talk on the telephone. The cat could be sleeping, lying on the bed.
"¢ The kitten lay on the bed. Its mother lay on the bed near him. They were lying on the bed together.
You might say "lie in the bed," but this would be highly unlikely. This bed describes not one's usual bed, but another bed, as in these few examples from the New York Times:
"¢ ..on Married Love," edited by Laura Chester (HarperCollins). When you know somebody that deeply, so many bodies lie in the bed alongside. We need a king-sized bed to contain them. Our bed is full of farmland, 200 acres including the back...
"¢ ...his news conference dissolved into a counselor's couch. ''I can't care about that,'' he said. ''We all lie in the bed we make. That's the life I lived. ''I'm happy that I'm 40 years old and I'm at peace with myself. It took...
In general, Joyce, "lie in bed" is the combination of words we use. In your sentence, poor Joanna may not actually be lying under the covers, but on top of them.
And, when there is a modifier of "bed" – in this case "her," – it is more permissible to change the expression a little. Joanna could "lie in bed" or "lie in her bed", or "lie on her bed" – but never "lie on bed." "Lie on bed" is not an idiomatic phrase.