Run the experiment:
For entertainment—try different levels of water flow with your existing speaker setup and see if there’s any overlap between the range of “audible entertainment” and “acceptable cooling.” The experiment is fast and cheap. Edit: You may not be able to find the right level of cooling without first doing some exercise nearby to heat yourself up.
For safety—Assuming you’ve found an optimal level of water flow, try dancing at various levels of intensity with a friend present in the bathroom to catch-you/call-an-ambulance/help in case of severe accident. Not quite as fast as the first experiment, but contingent on it and still free and relatively easy—plus amusing for a friend
Dance in the shower?
Even fairly restrained dancing over an extended period can elevate your heart-rate and trigger many of the benefits of exercise.
If you have a shower and live in an area where cold water is provided for free, then there is no cost. Additionally, this should address your sweat issue much like swimming in a pool.
It is indoors which eliminates sunlight.
Vis-a-vie the boredom constraint. Dancing to music by itself may be a varied enough activity to keep you mentally engaged. If this is insufficient you might consider audiobooks or talk radio.
Complications:
Safety. If you dance too vigorously then you may slip or injure yourself. You will know better whether this is a very likely issue. If it is, you may be able to mitigate with protective clothing (either purchased such as no slip water-shoes/socks or made from household objects; feasibility depending on your particular budget constraint and safety concerns).
You may not have a sufficiently strong set of external speakers to overcome the noise of the shower. If so, you might mitigate by (1) reducing the water flow for your shower, (2) purchasing or building louder speaks (for example, build a cone out of cardboard to amplify and direct the volume.)
Just cold water may be too cold for you. To mitigate either (1) add hot water (and possible add cost), (2) exercise outside the shower first to raise your body temperature, (3) acclimate yourself to colder water (this has been done by many people in the past either by necessity or due to a specific purpose such as Channel swimming).