Since this is homework, I'm going to just give some signposts. There are a lot of articles on curve flows, which have many similar computations in them. They may help. Likely the best reference (certainly the most seminal) is Gage and Hamilton, but there is a huge literature on curve flows these days. (Shameless plug: a new article of mine which is available on the arxiv might even help!)
I'm only going to answer the question 'show the length is preserved'. To prove the flow converges to a circle is a much more complicated story, and one I can not believe would be posed as homework.
Let's use the notation from your question, and take $C:I\times[0,T)\rightarrow\mathbb{R}^2$ to be the evolving family of curves. The length of each curve is defined as $ L(C(\cdot,t)) = \int_I |\partial_uC|\,du\,. $ To solve your problem, complete the following.
Step 1: Compute $\partial_t\partial_uC$.
Step 2: Compute $\partial_tL(C(\cdot,t))$.
Step 3: Use integration by parts (recall that $C$ is closed) to simplify the answer from step 2. What do you notice?