"Indeed. What sort of skills could you employ in my service, Gytha?"

"I am trained as a frontline combatant, my Thane. I have heard that you and your honorable companion favor ranged or magic options; I would be able to hold enemies in place and to draw their attacks towards me. I am very strong, and I can carry a lot beyond my armor and weapons. I am trained in the management and defense of a noble estate, and in the laws and customs of Skyrim's holds, and in those laws and customs relevant to trade or defense of our closest neighbors. I am trained as a bodyguard, and as a leader of soldiers, and I have served in the honorable Guard of Whiterun Hold."

"That all sounds like extremely useful experience. One more question. Do you always speak so formally?"

"...I can attempt to speak less formally if it is your wish, my Thane? But you are - exalted, and hopefully to be the one I owe homage to. It is proper to show my respect."

"Oh, I like this one."

"I do too."

Blush! "Thank you, my Thane."

"Then I think I've made my decision."

"I am honored, my Thane."

"Welcome to my service, Housecarl Gytha."

"Thank you for allowing me to serve you, Thane Elerki."

Smile.

"Are there any more formal ceremonies to go through?"

"You should inform the Jarl of your choice, my Thane, but - unless your culture demands it, no formal swearing ceremony is required. Though, if you want one..."

"You would be more than eager to oblige me?"

"I would, my Thane."

"Then kneel, housecarl, and swear to me your homage."

She does so, eagerly. 

Elerki liiiiiiiikes.

Then to see the Jarl.

"Gyða is a good choice," he says, approvingly. "May she serve you well."

"I trust that she will. Thank you for your hospitality, Jarl Balgruuf."

"The honor has been mine, Thane Elerki."

And time to go?

At least back to their suite to get ready. 

Right. And Gytha should get her things too.

She will meet them at their suite within the hour.