But code does get added! Duplicate copies of genes arise through replication errors. Transposons spread extra copies of themselves throughout the genome. Viruses attach their own code. Extra copies of entire chromosomes can be added.
But extra code is not always necessary to create a radically different creature. The number of genes in a man or a mouse are not so very different. The genes a simple animal has are enough to produce a gigantically large number of wildly different species, if exploited to their full potential.
And finally, there do exist "genetic toolkits" in nature that are capable of making highly complex, coordinated changes to a large number of genes at a single throw. (I recommend The Wisdom of the Genes by Christopher Wills for a description of some examples of this.) The ability to evolve does itself evolve.
So, do you believe that mouse DNA contains enough info to make a man? I think there are 3 billion polypeptides in human DNA so that would seem like a lot more than needed to make a mouse.
Hogwash.