The problem word here isn't "into", but rather "within" - it just doesn't fit here.
Inexact time: "Within {duration} of {point in time}"
"Within ... of" is used to state that something is close to something else. Take this example:
- "the second runner finished within two minutes of the winner's time"
You're stating the upper limit of the difference ("two minutes"), and the reference time ("the winner's time"), and so we understand that the difference between first and second place is some value that is less than two minutes. What you haven't told us is the exact time that the second runner finished.
Exact time: "{duration} into {period of time}"
This second form is slightly different: you're specifying an instant of time by reference to how far into a longer period it occurred.
- Two hours and five minutes into the scheduled time, the winning runner had already finished.
Here, you're stating the time something happened, not a difference in times. We don't know the exact date and time that the winner finished, but we know that it was 2:05' after the race started. It's a single instant.
The decision was taken at a point in time: use "into"
Look at the two possible "right" sentences from your example:
"within a week into of his nine-month tenure": This doesn't make sense. A tenure isn't a single moment in time, so you can't pin-point the time of another event relative to it. (to fix this, say "within a week of his nine-month tenure beginning", but this is clumsy)
"within a week into his nine-month tenure": The tenure lasts nine months, and the event happened around seven days after the tenure started.