In the late afternoon, I happened to see that you had an empty slot in your lesson schedule, although it was fully booked this morning.
I thought luck was on my side as I booked it.
ご質問ありがとうございます。
"In the late afternoon, I happened to see that you had an empty slot in your lesson schedule,"=「夕方、あなたのレッスンスケジュールに一つ枠が空いているのをたまたま見ました。」
"although it was fully booked this morning."=「今朝は完全に埋まっていたのに。」
"I thought luck was on my side as I booked it."=「今日は運がいい、と思いながら予約しました。」
ご参考に。