'Whatever' means 'whichever thing/job' etc. 'Whichever' would only work if followed by another noun.
For example, "Then you can be refreshed enough to start WHICHEVER JOB is next on your to-do list."
Alternatively, you could say 'whichever task' or 'whichever chore', or another noun with a similar meaning. 'Whichever' doesn't work on its own - we need to know 'whichever...what?'.