You can define multiple measures for the quality of your mesh. For example, gmsh uses a criterion that measures the element shape. It is based on the element radii ratio, i.e., the ratio between the inscribed and circumscribed circles (See reference 1). I think that you can use the Jacobian to describe the quality of elements as well, although you might want to check reference 2.
Regarding the Jacobian, the first thing to mention is that is a function of the element coordinates in higher-order elements. Then, for those elements, you would need to compute a figure o merit per element, e.g., the average of the determinant or the maybe the norm. In the case of linear elements, you will have a transformation like the one described in the following image

For this simple case the transformation is given by
$$\begin{pmatrix}x\\ y \end{pmatrix} = \mathbf{T}\begin{pmatrix}r\\ s \end{pmatrix} \equiv [J]\begin{pmatrix}r\\ s \end{pmatrix} + \begin{pmatrix}x_A\\ y_A \end{pmatrix} \enspace ,$$
with
$$[J] = \begin{bmatrix} x_B - x_A &x_C - x_A\\ y_B - y_A &y_C - y_A \end{bmatrix} \enspace ,$$
and $\det J = (x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A)$. That is the simplest way to think about the transformation, and you can just find it solving a linear system of equations, and it just depends on the position of your nodes.
For your problem, you would like to compute the Jacobian then apply the displacement to each node, and compute it one more time.
References
- Geuzaine, Christophe, and Jean‐François Remacle. "Gmsh: A 3‐D finite element mesh generator with built‐in pre‐and post‐processing facilities." International Journal for Numerical Methods in Engineering 79.11 (2009): 1309-1331.
- Pébay, Philippe, and Timothy Baker. "Analysis of triangle quality measures." Mathematics of Computation 72.244 (2003): 1817-1839.