I had to wait about 10 minutes for the anesthetic to kick in before my surgery.
この場合、次のような言い方ができますよ。
ーI had to wait about 10 minutes for the anesthetic to kick in before my surgery.
「手術前に麻酔が効くまで10分ほど待たなければいけなかった」
to wait about 10 minutes「10分ほど待つ」
anesthetic「麻酔」
to kick in「効く」
surgery「手術」