Paula i know you mentioned taxi to the hotel but have you considered the train, this goes direct into the airport and the sky tunnel link so it's really convenient for the hotel
with a family a friends railcard a family of four can travel return for approx £140 using a website like the trainline.com (booking 3 months in advance)
we used the train last trip and found it a suprisngly relaxing start to the trip, you can prebook a table seat and even plug in the laptop so my teen was over the moon