How can I create the equivalent of a fresh cake yeast from my sourdough culture?
Food Grade Yeasts, Food Technol. Biotechnol. 44 (3) 407–415 (2006) describes the process by which yeast is commercially prepared in good detail for an overview.

There are several stages of cultivation and growing and increasing the yeast biomass, culminating as its final steps:


  Treatments and packaging The yeast in the final trade bioreactor is
  concentrated by centrifugation and finally harvested by a filter press
  or a rotary vacuum filter, until it contains 27–33 % of dry cell mass.
  The yeast cake is blended with suitable amounts of water and
  emulsifiers and cutting oils (soybean or cottonseed oil) to obtain its
  extrudable form. The yeast is then packaged and shipped as compressed
  fresh baker’s yeast, or thermolysed and dried to form various types of
  dry yeast. The dried yeast is packed under vacuum or nitrogen
  atmosphere. The packaging method varies among manufacturers and
  depends on the type of yeast product.


Note that this process requires assorted items of industrial or laboratory equipment which may not be suitable to the home environment, including centrifuges and filter presses or vacuuum filters.

If you really only want to preserve your starter strain without maintaining an active starter, simply drying a sample and then storing it in the freezer may be a better option.  This process is described at Breadtopia.