If you want to solve the Laplace equation using the Finite Element method, there are 4 steps:
1) Discretize the domain (done by Triangle.NET)
2) Assemble the system matrix
3) Solve the linear system (use some math library like MathNet or CSparse.NET)
4) Check the solution, visualize the results (you can use Triangle.NETs RenderManager for basic visualization)
So the only thing you need to do, is to code the assembling routine. It's a great learning experience, so I suggest you have a go.
Here's a screenshot from a test app I wrote a while ago. It solves the laplace equation for a simple potential problem:
laplace(u)=0, u=0 on the outer boundary, u=1 and u=1 on the slits
It's less than 200 lines of code.
