Why can’t I start “Under New Management?”

Windhelm, Markarth, Solitude and Whiterun all have "influence" quests that each become available once you've done 5 of the randomly-generated jobs from Vex and Delvin in that city. Since jobs in Riften or cities you've already completed do not help your progress towards completing the influence quest, you can save before taking a job from Vex or Delvin and then reload until you get a city you haven't completed yet, if you feel so inclined.

Completing the influence quests unlocks fences in Markarth and Windhelm, but most importantly, each influence quest you complete increases the available gold pool of all fences in the world, up to 4000 once all quests are complete.

When you've completed all four influence quests, plus the main Thieves Guild storyline, you can become the guild master, receiving a significantly-upgraded suit of the guild armor you gained when joining.

First, complete the Thieves Guild quest line (ends with you returning the Skeleton Key).

Then you need to do Delvin and Vex's generic (radiant) jobs until you have done five (5) radiant quests for each of these cities:

• Markarth
• Windhelm
• Solitude
• Whiterun

It doesn't matter which kind of jobs you do for them. After you have completed five jobs in one city area, you will receive a special quest from Delvin for that area. Complete that quest to "upgrade" the Thieves Guild to a higher restoration status. Riften quests do not count. I would advise abandoning the Riften jobs unless you just want easy money. After you complete five jobs for one city, further jobs in that city won't progress the restoration. To make sure you only get jobs for cities you need, you can take a job from Vex/Delvin; then quicksave after you've ended dialogue with them, but before the job itself appears. If the job that you are given is for a city you do not need, simply load the quicksave and a new radiant quest will be assigned for you. This is easier and more efficient than simply abandoning quests, which also has been known to affect your "quests completed" count in the game.

In summary: 5 radiant quests for each city, and 4 special quests, one for each city, for a total of 24 quests.