Burd spends all of his time within Bruma Castle.

Where is Jearl?

Go to the runestone located at the base of the hill near the Cloud Ruler Temple at dusk. Jearl, a Mythic Dawn Agent, will be there and will attack you. Kill her and take the Basement Key and Jearl's Key off of her dead body. You'll also find the spy Saveri Faram at this location and time.

Who is staying with Jearl?

As she's very busy during the night, Jearl sleeps late and gets out of bed no earlier than 12pm. She spends the next two hours inside the house, talking with her partner-in-crime Saveri, until sitting down at the dinner table to take her lunch at 2pm.