Android - Is there a way to make a page in the Kindle app less brighter than the options available within the app?

You can change the brightness in the Kindle app by touching the "Aa" icon near the upper right. This will bring up the font, size and brightness settings, which you can then adjust to your liking.

Amazon Kindle brightness settings

To break the upper/lower limits of device settings, you'd most likely need to tweak your ROM, which requires the device to be rooted. I know there are some solutions for volume limits around ("booster"), so I won't rule out they do exist here as well.

To work around system settings at the "lower end", as it is your intention, you might e.g. want to take a look at Screen Filter:

Applies a shade that acts as a dimmer to ensure your eyes don't hurt. Far more powerful than Android's built-in brightness setting. Great for low-light gaming, web browsing, and eBook reading. It even saves battery life for AMOLED displays!

So this app uses a trick by adding a semi-transparent layer on top of what's displayed, which makes the display appear "darker".

Then there's also Night Mode, which has a similar goal (and uses a similar method to reach it):

Night Mode's main goal is to reduce the brightness of your screen lower than you can achieve with the default settings.

This application applies an overlay filter that acts as a dimmer to darken the screen and ensure your eyes don't hurt.

Pick your choice -- or check for another. Where there are two, there might be more (Super Dimmer, Screen Dimmer, Screen Filter, ...). A Playstore search will be your friend here (note that that's also all what I did -- I didn't use any of those solutions myself, so no guarantees :)