Image problem with the sheet not grounded

It doesn't matter if the plate is grounded or not, the field outside the plate is the same either way. You know this because the only constraint on that field is that its parallel component at the plate must be zero. That suffices to specify the image-charge field, as long as there is no net charge density on the infinite plate. If there was a net charge density on an ungrounded plate, that just adds a constant field perpendicular to the plate, but if there's no net charge density (i.e., no infinite charge on the plate), whether the plate is grounded or not will simply result in a negligible charge distributed over the whole infinite plate. In short, pulling charge from the ground simply distributes that charge over the whole infinite plate, with negligible effect.