There is an awful lot of confusion on this thread. You are not paying your daughter's Council Tax OP, you, as the owner occupier are paying your Council Tax.
If there is a single adult for whom this address is their sole or main residence (using the legal definition of this not some random one that's been dreamed up) then you will be entitled to SPD. The Council's contention seems to be that although your daughter isn't at your address at the moment she has gone travelling (I.e. is on holiday) and will return to your address afterwards. As she has no other address to give as a forwarding address this is a reasonable assumption. Your contention is that she has moved out permanently and will not return.
Showing the Council a flight ticket is meaningless. People go on holiday all the time, they don't get a CT discount for it!
If your home is still your daughters legal residence then you will not be able to get a single person discount until her legal residence changes. If you cannot get SPD you should apply for second adult rebate instead (as your daughter is on a low income). Second Adult Rebate is also 25% of the bill. As I said earlier, from the info. you have given you are entitled to either SPD or 2AR.
You need to speak to the Council - no-one else can fix this but them.
Also, if a property is empty and unoccupied it attracts a class C exemption for upto 3 months. To be empty and unoccupied it has to be unfurnished as well as unoccupied. This is why many rental properties do not qualify for it between tenants. Also if you purchase a property which has been empty and unoccupied and already used the 3 month exemption you will not be entitled to a new 3 months. The exemption is on the property not the liable party.
Council's have discretion on empty property and can charge 50% of the full rate or anything up to twice the full rate. It's to try and avoid the situation of people leaving property unoccupied for years and to get the property back into occupation. It's also to directly target holiday and second home owners.
Honesty OP, the easiest way to deal with this is to tell the council she has moved abroad and is not returning. If they insist on proof which you can't give then ask to apply for Second Adult Rebate until you or daughter can get the proof of change of address.
Years ago when I worked in CTax the office was populated with people who knew CT and rating law but now I swear council CT offices are populated by non-thinking automatons who just follow a script and don't apply and logic or empathy 
I'm sorry you are in this position, it must be very frustrating.