10 days doesn't appear to be sufficient. Even accounting for the problems caused by the fact she had to infiltrate the Tower to reach Egwene when she was propably expecting the biggest problem would be to get passed Sheriam, she reached Egwene much later than 10 days after she met Mat, and Mat noticed she still had numerous other letters to deliver (and so, people to find).

There was no chance Mat would agree to open the letter in ten days and do what it said, it's the same dilemna as telling him if he wanted to leave before 30 days he just had to open the letter and do what it said... No chance he would accept that (not only Mat, most people would choose to wait out the 30 days rather than risk getting caught in some AS trap). Verin wasn't taking a big risk there and yes, it seems she was hoping they'd settle on thirty days all along and he would open it after out of curiosity to know what crazy AS stuff he had just escaped, at what time if she was right not only would he be able to warn Elayne to close the Waygate but there would be a ta'veren with an formidable army stationned in Caemlyn.

Verin miscalculated because she didn't know Mat needed time in Caemlyn and was too happy to get this excuse to put off Thom a few more weeks. Had she known about the plans for Moiraine and the pressure Thom was putting on Mat, she would have realized there was a good chance she was screwed and she better arrange a plan B involving someone else than Mat.