Both Michael and Gilday were desperate men: Michael desperate to find absolution for his past sins; Gilday to escape blame for the Vatican Bank's peril.
TB, the Church's anointing Michael as a Papal Knight only washed away some of Michael's sins. It was an outward symbol of that washing of which the public would be aware. That helped him to some extent because of the accolades it would earn him. But Michael's guilt, aggravated by his diabetes, ran much deeper. Who knows for sure what he felt. However, money, power, and manipulation was all that he knew. Immobliere would give him the money and power which, in turn, would allow him to manipulate the absolution that he sought. Of course, no amount of manipulation would achieve that.