Ten days. The deal was 10 days and obey the instructions in the letter or else wait 30 days in Caemlyn if he chose not to open the letter.

Why the ten day's wait? If there's anything about the Shadow's plans in there, the wait is for the same reason she could not reveal the BA to Egwene until "the hour of her death": her Oaths. Verin may have trusted Mat's promise, or she may have set a ward on the letter that would dissolve after ten days.

Verin likely outwitted Mat. Apparently, Mat has decided not to open the letter after 10 days. By day 30, he will have met his promise to wait and would not be bound anymore to obey Verin's instructions if he read them. He would then see it as safe enough to open the letter, and no doubt he'll be curious.... I get the feeling that whatever is in there is stuff Mat might prefer not to have known but is also something he won't be able to overlook once he knows... and part of the instructions might tell him to remain in Caemlyn and explain why. Mat might have to decide between Moiraine's letter and Verin's letter (could be his two Aes Sedai on a balance scale, we don't know for sure this was resolved by deciding he had a debt to Teslyn over Joline). It would be even more suspenseful if Mat dithered until he's left Caemlyn for Whitebridge and Ghenjei until he opens the letter and learns he must stay in Caemlyn... for Elayne's sake, for instance, or Rand's sake. He doesn't know of Min's viewing about Moiraine and the importance to save her, but we do.

There's very likely instructions about how to get the Horn in there (I suspect Verin moved it when she went to the WT - I think she gave placed it in Laras' hiding place, for myself), but that's probably not the end of it. I suspect Verin left information about the beginning of the LB in there, and the importance of Mat being present when the Shadow strikes.

It's possible Verin left information on Taim to someone (perhaps Mat, but not necessarily - it could be Elayne, or someone in Rand's inner circle like Alanna Mosvani she could easily find - and I would think leaving a letter to Logain if she could find him is a strong possibility - he's got BA with him to deal with), but I doubt she would have set Mat the task of dealing with Taim. Taim will require the combined efforts of Egwene and Rand/Logain, especially if the remnants of the BA have gone to hide at the BT as seems possible.

What we know is that Verin encouraged Rand to dismiss Taim for now. She downplayed Logain's revelations about him. I think she knew it wasn't time to deal with him, that Rand could even precipitate the start of TG if he did.

Personally, I suspect Verin knew, suspected or has puzzled out that the Shadow will strike at Caemlyn first.

The big secret is who else Verin may have left letters to. Perrin? Elayne? Logain? Rand via someone else (eg: Alanna?), Cadsuane or Min via someone else (Alanna, Samitsu etc.?), a letter to Nynaeve (about Isam?) via someone else etc.