Yes your daughter is quite right.
She will probably clear �840 per month, maybe less.
This is an extremely good wage for a 19 year old after tax.
13,000 P.A = 1083.33 P.M
Tax allowance = � 5620 beable to earn before being taxed at 20%
Anything over � 468.33 P.M
Tax �123
Nation Ins approx 7.5% = � 86.64
Pension at 8.5 % = �81.20
Leaves your daughter with � 792.50 Approx P.M
However there is more to it than that because you get an allowance on your Pension and NI contributions.
So if you add � 50 back on it gives you � 842.50
Hope this helps !!!