I'de like to point out that Voldy is not a pureblood he is a half blood, his father was a muggle, and that Draco DID NOT KILL DUMBLEDORE! Snape did.
Now to mugglemathdork.
I like anAnachronism's idea about thinking he had something on Harry. Might be neat if Ron cought on, got mad and Harry still has no idea whats going on.
Draco has a lot of pride as well, what if circumstances demanded that he asked someone he considered bellow him for help? Like the cleaning scene you mentioned, he obviously has no idea how to do it, what if he had to struggle with the choice of asking the muggleborn Hermione for help or continuing the manual labor until his hands were covered in throbbing blisters.
Or he could be in a hurry to get somewhere on time and is desperate to leave but can't until its finished.
Whichever applies to your plot.
Harry knows how to do all that too but I think Daco would commit himself to the life of a house elf before he asked Potter for help. Hermiones not so intemidating, though thats not to say that she wouldn't have fun with it, like demand he say please or do something for her in return.
Hope this helped!