What can I deduce from a matrix by its Jordan Canonical Form? I solved an interesting, but troubling, problem stated as follows:

If $f(x) = (x-2)^3(x+7)^2 $ is the characteristic polynomial and $p(x) = (x-2)^2(x+7)$ is the minimal polynomial for a $5$x$5$ complex matrix $A$, what is the Jordan Form of $A$?

I found the Jordan form to be:
$$\begin{bmatrix}
2 & 0 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 & 0 \\
0 & 0 & 2 & 0 & 0 \\
0 & 0 & 0 & -7 & 0 \\
0 & 0 & 0 & 0 & -7 \\
\end{bmatrix}$$
Using a combination of Primary Decomposition Theorem, Cyclic Decomposition Theorem, and Caley Hamilton Theorem.
However, my concern is how I would use such a matrix. I don't know what basis I'm using to get this form, nor do I have handy the form of $A$. I believe I could extract some basic quantities like the rank and nullity of $A$ from here, but finding the image and kernel would be effectively useless without being able to convert the vectors to and from standard coordinates as far as I can tell.
I suppose my questions are:

What kind of information is the Jordan Form providing me with here? 
Can I, without first finding $A$, extract the basis that I am using (up to ordering) to represent $A$ in this Jordan form?


You can easily extract the following information about a matrix in JCF:
Its rank, its eigenvalues and their algebraic and geometric multiplicites,  its determinant, its trace ...
I think that's a lot of info that you cannot get easily from a matrix not in JCF. 
Now, basis with eigenvectors and/or generalized eigenvectors you cannot get it, as far as I can see, only from the JCF. In particular, when the matrix is not diagonalizable, as in your example, you'll have generalized eigenvectors (which are not eigenvectors).
BTW, I think most of the usual representations of JCF use the upper triangular form, which menas that $\;1\;$ you put there in entry 2-1 would rather go in entry 1-2.
