If she figures out how to glide in a consistently straight line, then she can leave the island behind quickly and had better turn back unless she wants to try a water landing.
Thanjen follows her down, after making some shape adjustments so he isn't flying faster than her all the time.
There's the beach again.
“Do you want to go up again, or try to take off from the ground?”
(Aaaargh.)
“—I really think we should be getting you equipped to communicate now. Then you can read entire books on how to be all kinds of things whenever you want, better than I can teach you.”
(Maybe he shouldn't have asked that. But she is at least safer now, and some things he was resolutely not thinking about are getting more insistent now.)
“I am birds all the time. I don't know that much about swimming creatures. I'm sure there are much better books on the subject.”
“If you're in a hurry, why not just try doing it? Then you'll know what you need to learn.”
She frowns at him, sighs, and then starts making herself a mermaid tail out of griffin parts that she thinks will be relatively easy to reconstruct later. When she has something that is appropriately shiny and scaly and fits over her legs well she makes a big globe of glass full of air and then attempts to compress it down.
Some discreetly frantic research later, he teaches her how to make a pressure regulator so she can safely breathe from it, and warns her not to dive deep with regular air.
“You need special materials and a particular machine to filter the air. And you also need to watch how fast you go down or up. It's complicated and I'm really not qualified to teach you how to do it safely.”
He dry-suit-snorkels along at the surface to keep an eye on her. Not that there are any too-deep places anywhere this close to the beach, but just on the general principle of who knows what she will want to do next.
She mermaids about quite happily and doesn't drown. She seems to find having human joints under the tail inconvenient, and at one point flops out of the water long enough to grab more material from the griffin shell and add it on so her tail is longer and she can get more propulsion without unsightly knee-bending.