Constructing Idempotent Generator of Idempotent Ideal
Solution 1:
One can reconstruct a method by considering the usual proof of Nakayama's lemma.
Suppose you know that the ideal is generated by $n$ elements, $(x_1, ..., x_n)$. By assumption, we may write $x_i = \sum a_{ij} x_j$, where the $a_{ij} \in I$. The element we're looking for is $p(1) -1$, where $p$ is the characteristic polynomial of the matrix $(a_{ij})$.
To see this, consult the proof of NAK in, say, Matsumura or Atiyah-Macdonald.