OK - as I understand it so far:
You DD has serious helath issues.
You rang the LA and asked if she should have a Statement and they said that Statements were only issued for learning difficulties and not in response to medical needs. They told you she would';t get one.
Basically the LA lied to you.
Here's what you can do:
School has a legal duty to identify those children who have 'special needs'. Those needs can be educational, social, medical or any other type of need that they have but which other children of a similar age don't have. The other thing is that the special need has to have existed for a while and will continue - which your DD's has and will do
So your DD's serious medical problems means she has a sepcial need.
School are supposed to identify what extra help any child like your DD needs and to provide that help.
The problem is that school may underestimate the help that's needed and say that they are meeting her needs when, in fact, they are not.
That's why use should ask for a statutory assesment to be carried out by the LA that may lead to the LA issuing a Statement that tells school how much need your daughter requires.
You must make this formal request for a Statement directly to the LA and in writing. This is very impportant as when you make that request in writing you strat a legal process - so the LA must consider your rqeuest and has certain timelines to make their decision. This is very different indeed to raining the LA and getting some isiot who doesn't know all the facts telling you that she won't qualify for a Statement.
The LA can only determine whether a Statement will be needed when they have taken advice from you, school, her doctors etc that will tell them exactly what help she needs. Until that's been done everyon is just guessing at what she needs.
Your first step should be to write a short letter to the LA saying what your daughter's difficulties are and asking the LA to make a Statutory assessment.
There's a link here to IPSEA webiste that shows you the sort of letter you need to write. You can enclose medical letters etc as evidence that your daughter does have medical problems.
How to Apply Formally for a Statement
But please do it soon because she needs help as soon as possible and also because Statements are being replaced soon with something else.
If you need to ask anything please come bacj and someone will be sure to help.