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.
(..) Oh wretched state, oh bosom black as death Oh limed soul, that struggling to be free an more engaged Help, angels, make assay. Bow, stubborn knees And heart with strings of steel, be soft as sinews of the new-born babe All may be well Now might I do it, now he is praying (..)2015 Hamlet
(..) - With what, in the name of heaven'? My lord, as I was sewing in my chamber Lord Hamlet, with his doublet all unbraced... Pale as his shirt, his knees mocking each other, and with a look so piteous in purport... ...as if he had been loosed out of hell to speak of horrors, he comes before me (..)2015 Hamlet