torture and drugs, that could work, she'd want to hurt him out of revenge and she'd have to keep him drugged and restrained so he doesn't get a chance to escape and whatever information about himself he lets slip when he's drugged out of his mind she could use against him if/when she feels that physical pain isn't effective enough...and he'd be especially vulnerable to that if it happens after the tunnels considering he's lost everything, he's still grieving and he's filled with raged but completely helpless to act on any of it; that's a lot of weak points to exploit.

I could see Claire being entirely motivated by revenge at first and slowly realising that she pretty much has power of life and death over Vladimir and she likes that. And she could rationalize it by thinking he deserves it for what he put her through and all the lives he ruined

Would be really interesting to add Stockholm syndrome to it, Vladimir would be completely dependent on her as long as he's injured/restrained/drugged so if Claire were to deny him essentials like food or water (or his next fix if she gets him addicted to something) she could condition him to obey her every command or accept whatever violence she dishes out with a 'please' and 'thank you'

....basically I think I might be beside you on that ride to Hell but damn I want this bad