He was forced to ask for their help 'on bended knee'. [=he was forced to beg for their help; he was forced to ask for their help in a very humble way]
If you want to clean the floor properly, you have to 'get down on your (hands and) knees' [=you have to kneel down on the floor] and start scrubbing.
She 'dropped/fell/sank to her knees' [=she knelt down] and begged for forgiveness. = She 'got down on her knees' and begged for forgiveness.
(..) And in order to forge a lasting bond with the Meereenese people, I will marry the leader of an ancient family. Thankfully a suitor is already on his knees. Don't worry. I'll be fine. Nothing broken, I don't think. Where are we now? Not the Rhoyne. (..)2015 TvShow: Game of Thrones Title: Kill the Boy Season: 5 Episode 5
(..) Forgive me. I didn't think-- Come here. You mustn't keep secrets from me, Reek. Get on your knees. Give me your hand. I forgive you. I trust you find your chamber suitable, my lady. Yes, thank you, (..)2015 TvShow: Game of Thrones Title: Kill the Boy Season: 5 Episode 5