I figured it out, here is an outline for those that might be interested
The idea I had worked perfectly well, we can show that
is dense in irrespective if is unital or not. The trick is in noticing that we can create matrices with either 1 (in the unital case) or the approximate identity (non-unital case) at the positions . The span of these matrices will be dense in .
The Cauchy sequences are also easy since we are dealing with a finite dimensional space . The inner products and induce well defined norms. Now there is a theorem which states that every finite dimensional normed space in complete.
All the conditions are then fullfilled and we see that is in fact an -Imprimitivity bimodule and hence and are Morita equivalent.