There are 3 ways to earn Kudosh.
-Complete career goals
The list of career goals and rewards can be found on the Hospital selection menu, if you presscn_X
-Complete in-game challenges
When in your hospital you can press the cn_up button to go to the list of your current challenges. Complete these to earn the rewards.
-Research
You can use the research room to generate 20Kudosh, this will cost $1,000 each time.