First of all, the acetic acid solvent is no longer anhydrous after the reaction, as the NaBO3·4H2O is ~40% water by weight, so you have at least 10% water in the acetic acid.
If you want to reclaim this 90% acid, I suggest that you distill off around 2000 mL of it under vacuum (as boiling the reaction mixture in 114°C acetic acid might lower yields), then dilute the residue with water and extract as in the writeup. The 90% acetic acid reclaimed will have to be dried before it is reused in the next run, and I'm not sure how successful the distillation from sulfuric acid has been proven to be, but the results are available in TFSE.
As hydrolysis of the P2P enol formate produced in the baeyer-villiger oxidation will give rise to formic acid, any acetic acid distilled off will contain formic acid as well, and this acid will decompose to carbon monoxide if distilled from sulfuric acid, so watch out for that happening.