How can I manually turn on do not disturb at random times like when taking a nap?

First, ensure that you have updated to the most recent version of iPadOS. At time of writing, iPadOS 16.6 is current.

Next, set-up the Do Not Disturb Focus mode:

Use Do Not Disturb with Focus on your iPhone or iPad - Apple Support

After you've set-up this Focus Mode, it can be manually enabled/disabled from the iPad Control Centre. To reveal the Control Centre, swipe-down from the top-right corner of the screen:

Additional information about Focus modes and the Control Centre can be found within the iPad User Guide:

Set up a Focus on iPad - Apple Support

Use and customise Control Center on iPad - Apple Support

Turn a Focus on or off on iPad - Apple Support

You will be able to do this using the Control Center. You can access this by swiping down from the top right corner of the screen by where you see the battery level. Once you pull that down, you’ll see “Focus” if you click on that, you’ll have the option to pick a different focus that you want. You will also be able to create a new one if you want and set it up to do what you want when that focus becomes active.