At long last, this week will be our last, and then we're taking a break for a while.
Literal
Finally this week [object-を] limit [with-に], for-a-while [genitive-の] honorific-break [copula-です].
Another ~を限りに example — 'with this week as the last.' いよいよ ('at last, finally, at long last') conveys mounting anticipation or finality. お休み is the bikago-honored form of 休み ('break, rest'), here used to describe the upcoming hiatus. Typical of a radio show or podcast sign-off.