Your daughter's passport remains valid up until the expiry date printed on it. (New applicants for passports are regarded as 'adults' from the age of 16, but a child passport, acquired before the age of 16, continues to be valid after the holder's 16th birthday, up until the normal expiry date).
The UK and Spanish authorities are unlikely to see any difficulties with the girls travelling together but it might be wisest to get your niece's parents to provide a letter stating that they agree to her travelling and that they will meet her on arrival in Tenerife. It's unlikely that your daughter will require such a letter but it would probably still be wisest to provide her with one.
However you must check the airline's specific requirements. (Each airline has their own rules). Some airlines will not allow the girls to travel. (For example, from 1st April this year, Ryanair won't carry any under-18s unless they're accompanied by an adult who has been booked on the same reservation). However most airlines permit under-16s to travel when accompanied by someone over 16. There is usually a requirement for a parent or guardian to complete an authorisation form at both departure airports.
Chris