Whilst you are getting her health checked out I would go to her, make no eye contact, don't speak to her or touch her, but let her out into the garden/ slip her lead on and take her out (whatever is appropriate) and give her chance to pee. You are facilitating her being able to get to pee - nothing else, so be neither friendly nor mean, completely neutral.
i find that not rewarding the Basenji with eye contact, praise etc it helps prevent the whining be rewarding by getting attention from you and thus increasing its frequency, but allows the animal to relieve itself if necessary. I've had a couple of times when this has happened with my boy - I've reacted instantly to his whines with taking him out (and when I was living in a 7th floor apt this was no easy feat) on each occasion he immediately pee'd and then went straight back into his crate to sleep for the rest of the night. To be honest, if he was comfy he'd hate going out into the cold night unless it was absolutely essential!