How much profile data should go in a JWT token?

You should put just enough information in your JWT token to authenticate your user without needing to make additional database requests. That's the whole point of the payload - so that authentication can be distributed. If you need to make database requests to authenticate every API call then you have not achieved distribution because you will still be centralizing your authentication process at the database.

However, do note that everything you put into your JWT tokens are public. Anyone can do a base64 decode on the token to extract the payload. Therefore I would suggest not including your user's email address in the payload if you don't use it often (for every API call). Malicious scripts may somehow get access to the token and harvest the email addresses (which does have value and can be sold to "email marketers"). The user's ID should be enough to identify the user.

If your UI needs to display the user's name then you can also include the name but personally I'd delegate that to a profile API call (I normally implement a /myself endpoint for this).

I'd also include the user's role in the payload to avoid needing to query the database to check if the user has permission to make an API call.