I would also type up a timeline, with dates and all relevant information about each incident (being as unbiased and scientifically dispassionate as possible). People tend to take things more seriously when presented with something physical that they can't dismiss. If you are concerned that they won't believe you (as has happened to me), list witnesses or people who can confirm each incident. Then attach to the back of the timeline all relevant emails and other written messages she has sent you, with the important parts (parts that show bullying) highlighted, as well as your responses telling her to stop.
Boss: she has your best interests in mind.
You: Even if she did, she still did UVW as you can see here, which is unacceptable according to company policy. Her actions are what I have a problem with, not her interests.
Boss: but I'm sure she meant well.
You: Perhaps, but she still did XYZ as you can see here, which is unacceptable according to company policy. Her actions are what I have a problem with, not her intentions.
etc. until Boss stops skirting the issue.
Go to HR first, with the typed list. Make it clear what you want, also: Jane to be told to stop bullying you (and it is bullying: you'll want to use that word repeatedly), stop eavesdropping in the workplace, stop gossiping and spreading rumours about what she has been told is sensitive, private, and none of her business, and so on. Make another list if it helps you brainstorm and get your thoughts in order. Don't think about if something will upset Jane: be coldly professional. You want her to do her job professionally, not to disrupt yours and the workplace with her bullying. If she has a problem with that, she's being unreasonable, and there's nothing you can do.
The gym advice is very good. The next time she signs you up for something or yells unkind things at you, tell her once firmly to stop (if you haven't already). Then, when she repeats the offence, go to gym management and explain that there is a gym member who is shouting obscene things at you and/or signing you up for classes without your permission or knowledge, and that she persists despite you telling her to stop. The gym is unusable as long as this continues. Either they deal with it or refund you the money for a service they are no longer providing.