We study the dynamics of 2D inextensible immersed interface problems. These problems consider a closed and inextensible filament with no thinckness immersed in 2D Stokes flow. Through the boundary integral method, we reformulate the problem into two contour equations, an evolution equation and a tension determination equation. The tension determination equation is for determining the tension on the filament that ensures the filament inextensibility. We first study the well-posedness and the regularity of the generalized tension determination problem in Holder spaces. Next, we use a suitable time-weighted Holder space to study the well-posedness and the regularity of the dynamic problem.