Runner the only legal tests you have to satisfy for an EHCNA are a) has or may have SEN, and b) may need SEN provision to be made via an EHCP. Anything else is unlawful. LIFT is unique to your area, other LAs call their services different things - just to confuse parents further, I'm sure.
It is frustrating when schools pass the buck and state there's nothing they can do. They can refer for medical needs tuition, apply for an EHCNA, apply for high needs top up funding, ask the specialist teaching service for advice, support a CAMHS or local universal services referral. If a child can go on site, or even if not in this virtual world we have now, they can provide mentoring, counselling, social skills support, emotional literacy support, 1:1.
no one offers anything unless you push is spot on. If there's ever a time to be a pushy parent now is it.
Have you looked to see if you are eligible for a Family Fund Grant?
Muddling SARs will show you what information others hold on DD. Even the GP notes should highlight DD's needs - presumably you discussed DD's difficulties to get a CAMHS referral. Once you get the SARs data cross check them to ensure you have been given everything. For example, an email between the school and LA should be in both.
IPSEA will be able to direct you to EPs near you accepting tribunal work. Starting a thread on the SN boards may work too. You don't need evidence of a gradual decline. Technically you don't even need the school to have done anything, case law shows it's possible to get an EHCNA when the school could do more but won't. Although it is obviously easier if support has already been tried. It's also possible to argue DD's needs can't be met without an assessment because there is insufficient awareness of her needs and the provision she requires. As DD was receiving medical needs tuition and it would continue if she hadn't aged out it has been accepted DD is too ill to attend school so you already have a strong case. As a starting point look at this.
When you phone DWP ask to become DD's appointee. For PIP reminders and prompting count. I imagine you remind/prompt/support DD to eat/drink, take medication, bathe, dress, prepare a meal, mix with others, planning and following a journey, potentially also meeting the communicating and budgeting criteria too. Unfortunately, shopping isn't considered under preparing a meal, although one of the mobility questions considers planning and following a journey and needing support to go out.
Don't state every day is a bad day, DWP know conditions vary and except for a couple of exceptions on safety grounds you only have to satisfy the criteria more than 50% of the time for them to apply. Also, if you are called for a F2F (some have now resumed) and you are having a better day it undermines your whole claim.
If DD registers with your local careers service there is a 20 week run on of child benefit. Otherwise for prescriptions/dentist etc. look in to a HC1 form. Failing that a pre payment certificate may save money.