A wee bit amused by some of the "oh, your poor employer" wringing of hands in some comments. In some industries and locations, when job offers come up, you have to snatch them with both hands, however inconvenient that might be for business owners who haven't managed to get proper procedures in place and can't separate personal feelings from pure business like some of the people I worked with in my 20s and 30s (mainly in SMEs).
On this occasion the OP is leaving for work purposes; would some of the comments she's had be less harsh if she were leaving suddenly for illness, or to look after a relative? And then in some parts of the civil service and other industries, when you apply for a new job, the referencing and security vetting can take up to a year so you might not always know when a firm job offer is going to be sent to you.
And it was a (pricey) conference, not a course. At presumably £1K per day per delegate (assuming 4 days) that's some conference - I don't even want to guess what field of work it's in or what the location was! You can almost do some proper postgrad qualifications for that, with letters after your name, and a certificate, and a floppy felt hat and gown and everything at the end....
I know you've said that the Office Manager confirmed the money was paid but I'd be tempted to double check with the company; I also have a strong suspicion, as someone else said, that your company would probably have been penalised anyway if they hadn't sent someone else on that course.
If your current company has a bit of a problem with people leaving suddenly and regularly, and high levels of sickness, they might want to look at why?
I'm loving this world where people can predict the future so they would never suddenly leave a job where they'd been sent on a training course in the recent past - can I have one of them there crystal balls please?!
Oh, and as someone else said, Human Remains (sorry, any HR people, I know some of you ARE very good, but really good HR people are as rare as hen's teeth in my experience...) are NOT on your side. They are there to enforce the employer's point of view and try to make sure aforesaid employer does not end up in front of a tribunal. And my understanding is that they did not make you sign any agreement to pay the money back in the event of you leaving within a certain timeframe.
On a serious note, I know lots of people (including me) have suggested ACAS, and in my experience they really are pretty helpful.