Maybe the laptop is powering on, but the screen is dark because of no backlight. After waiting a minute after powering on the laptop press the Caps Lock key. Does it light up? If so, then the laptop is booting and you may have a bad Display Assembly.
Sometimes if you close the display and open it very slow & careful the display backlight will light up. Try only opening the display an inch or two to see if the display lights up & stays lit which will confirm a faulty Display Assembly. If you open up the Display all the way you may find the display will go dark.
If the issue is with the display backlight, then you may be eligible for a free Apple display repair:
https://support.apple.com/13-inch-macbook-pro-display-backlight-service
Edit: It may also be eligible for other free Apple repair programs such as one for the battery:
https://support.apple.com/13inch-macbookpro-battery-replacement
Here is the current list of all publicly acknowledged free repair programs for various Apple products (some have multiple programs):
https://support.apple.com/exchange_repair