Correct.
Now that you've cleared up the dates and the vagueness of his oral notice, yes, that's correct (except for the 28 day thing, which I'll touch on later).
Let's call April 17 the surrender date and assume (you didn't say) that he paid April's rent on April 1.
Now let's look at Paragraph 6:
"(6) Tenant moving out without notice. If any periodic tenant vacates the premises without notice to the landlord and fails to pay rent when due for any period, such tenancy is terminated as of the first date on which it would have terminated had the landlord been given proper notice on the day the landlord learns of the removal."
Which has to be interpreted in conjunction with Paragraph 2(b):
"1. A periodic tenancy can be terminated by notice under this section only at the end of a rental period."
See if Mr K agrees with this:
For notice to be proper, the tenant would have had to give you written notice at least 28 days prior to the end of April to terminate the tenancy on April 30 and he would have had to pay April's rent and he would not be entitled to a partial refund if he left prior to April 30.
Moving on to the move out date of April 17 without proper notice. The 28 day notice thing doesn't apply to that because proper notice given on April 17 would have been for a move out date of May 31 and he would have had to pay you for the Month of May and not be entitled to a refund for May either.
Now that you've clarified the dates and the lack of notice, it actually appears that he would owe you for the full month of May in addition to no refund for the month of April.
Two things:
1 - Paragraph 6 applies if he doesn't pay you for any required rent period. You will have to wait until after May 1 (if that's the rent due date) before you can pursue May's rent in court.
2 - There is still the matter of mitigation. I suggest you read 704.29 very carefully. My take on that is you haven't "expressly agreed to accept a surrender of the premises and end the tenant's liability" because he left the keys while you weren't home and you have continuously insisted that he still owes more rent. The problem with that is that you have to start advertising for a replacement roommate now and get one as soon as reasonably possible. The longer you wait to start mitigating the more likely that he will be successful in raising the defense of lack of mitigation and reduce or eliminate your ability to collect any additional rent. In fact, if you decide not to replace him then my interpretation is that you would owe him a refund for the period from April 18 to April 30. By not mitigating you will have relieved him of liability as of his move out date.
Yes, from a practical standpoint, it would appear prudent to get out from under this as cheaply and as quickly as possible.
I faced exactly that option many times during my 20 years owning rental properties.

