In reply to JJL:
Charges for non attendance at conferences are common in my industry. I have no idea how enforceable they are although in general, a Credit Card is required at time of booking so that puts the organiser in a better place than yours who is trying to get you to physically pay up (rather than just debiting the card supplied).
However - on a reputational and relationship basis I've found that the conferences can be cancelled at the last minute without issue. Indeed they are generally so keen to sign up attendees that 6 months ahead when they are trying to get your name on list, if you raise cancelation/last minute changes in plans as a concern and reason you don't want to commit they generally say that's fine, just let us know etc.
So - The practice is common, I've never had a problem cancelling on short notice when I need to and I'd imagine simply having a word, explaining it was an unavoidable medical emergency and surely as they will want you and colleagues to attend in future they are not going to be daft enough to actually enforce the charge (and have you and your firm never ever attend one of their events again) will probably be enough to make it go away. Along with a few comments about how charging you would damage their reputation and make people less likely to commit in future.
If that doesn't work get legal input (which I'm afraid I cant supply) - I do suspect that commercial common sense will prevail if you handle the situation right. Also, in my office, if I commit to a work conference my expenses to attend would be paid by employer and that would include cancelation - my employer wouldn't expect me to be personally liable for a work conference... so I hope your company will step in and either pay or provide legal expertise - its a work issue, not a personal one surely.