There’s probably more than one way to solve this. Here’s how I would go about it.
From the intersecting chords theorem, you know that CPxDP = 4x10. You also know that CP+DP=13 (=CD). You now have two equations involving CP and DP.
Combining the two equations gives you a quadratic expression which when solved gives you CP= 5 and DP=8.
The distance from the midpoint of CD to P can be found to be 1.5.
The distance from the midpoint of AB to P can be found to be 3.0.
The distance of P to the center of the circle is the same as the diagonal of a rectangle of sides 1.5 and 3.0 which can be evaluated from the Pythagorean theorem (approximately 3.354).