Jauffre is afraid that just Burd's guards won't be enough to defend Bruma in the coming battle. The city needs as much allies as it can get. Time to talk to other cities' counts and try to persuade them to send reinforcements. It won't be easy, though, as most of them are troubled by their very own Oblivion gates., which will have to be closed. Guess who's up to the task.
This quest is in fact optional. You don't have to close all the gates, you don't even have to close any one of them, but keep in mind that the battle for Bruma won't be a walk in the park and some help would be useful.
Go to palace in Imperial City and find chancellor Ocato. He'll refuse to send any help and will tell you to pay each city an individual visit. You can't do anything about him right now.
When you close the Kvatch gate, Savlian Matius will ask you for further help in taking the city back. Agree and follow him. After defeating some monsters talk to him and go back to the Chapel to talk to Berich Inian. Follow him through the underground pass. Eventually you'll reach a wheel that opens the castle courtyard gate, use it.
You'll get some backup from Savlian and his men now. Talk to him and go inside the castle. You have to find a prince. Turns out you were too late after all, as he lays dead on the floor. Take his royal ring and give it to Matius, who'll give you his cuirass in return. He'll also agree to send some help to Bruma.
First you have to find and enter the Oblivion gate north of the city. Once you're in there, go forwards to see a bridge and a tower on the left. Inside you can get some stuff from the body of a Mythic Dawn cultist. Go further and enter Nether Tunnels. After going through some simple dungeon levels, you'll end up outdoors again. Go to Red Gnash Tunnels. This time it's shorter. When you are outside yet again, follow the path to reach The Tower Portal. Enter doors in that order: Rending Halls, Flesh Spire, dead Halls, Flesh Spire, Corridors of Dark Salvation, Sigillum Sanguis. There's another Sigil Stone in there. Touch it to close the gate.
When you do this, go to Countess Umbranox in Castle Anvil and tell her about your deed.
HINT: You'll have to do pretty much the same in Chorrol, Cheydinhal, Leyawiin, Bravil and Skingrad. The procedure of gate-closing is identical as in Anvil, Bruma and Kvatch. It's always about getting to Sigillum Sanguis and touching the Sigil Stone.