No, it won't be enough for the doctor's letter to say "Afsana has had PND and the family requires support".
What appeals panels look for, when the parent mentions that there is a particular social or medical need for the child to have a place at the preferred school, is confirmation from a health care professional (or social worker, where appropriate) that that is the case and that the preferred school is better able to meet those needs than any other school. A doctor's letter that mentions the health problems but does not say that in that doctor's professional opinion the child needs a place at that one particular school is not enough.
As clam says, you need also to check whether your LEA takes account of parents' needs or only those of the child. If the admissions category for social and medical need only covers the child's needs, you will need also to spell out the implications of your health issues for your child.
Is this an infant class size appeal? If it is, and you are raising the issues about your health for the first time now, it is unlikely that the panel will allow your appeal, as the LEA can't be blamed for not acting on information it didn't have. You need to ask the LEA how you can get your application re-categorised as social/medical need and therefore get moved into a higher priority group on the waiting list. Most LEAs have a panel (not the appeal panel) that adjudicates on this.
Lastly, nobody will think you are a bad mother. People on the panel are quite likely to know about PND (they may even have had it). They will be sympathetic to your situation. Even so, there are very limited circumstances in which panels can allow an ICS appeal - as I'm sure you've already been told, they will be looking for evidence of an error which deprived your child of a place, admission arrangements that are contrary to law or the admissions code or a decision that is so unreasonable it is immoral or perverse.