Showers are available at the public beaches located at both the Camp Branch and Little Platte public use areas.
I just found a lake map
http://www.nwk.usace.army.mil/smithville/smithville_b.html Scroll to the bottom of the page to see the roads and lake facilities. It could be clearer but you can find both of the areas mentioned.
If you've not located Heartland Cove look for county route W on the west side of the lake leading off of US 169 between 188th and 208th street. Follow W east 1/4 mile to the first left, (marked as route F) take that about 1/2 mile to King Road and follow King to the right, to the sharp left turn before it dead ends in a turnaround and public fishing access location. If you reach the turnaround -- just come back 1/2 mile and you'll be back at the wire rope gated entrance that leads to the Cove - it's about a 7 minute walk from the gravel road to the Cove.
I notice that the street sign for King Road is down and missing. So no reference there anymore. There is a small inconspicuous "H over C" stake at the wire rope entrance to the field leading to the Cove. That comes and goes - gets tossed into the weeds once or twice a season and I resurrect it.
If someone would like to scout out other secluded coves that'd be great and the map can help. I want to try the other side of King Road - it's split by the lake -- but you can come at the other side of it from the north off of 208th street and that might be worth checking out sometime.
Short answer on the showers - either public beach but during prime season it probably costs a lake entrance fee to get to them.
Dick