You can disable sleep mode and screensaver mode using shell commands via ADB, this works for the 4k Firestick and should work for the previous model although I would suggest googling first as what I’m about to provide you with is how I did it on mine and family members 4k firesticks.
Anyway you need to get an app called ADBLink, it’s available on linux, mac OS and windows, best thing about this app, is that as long as you’re connected to the same network, be it wifi or ethernet, you can still use this 
In your firestick you want to enable ADB debugging, by gonig to Settings > My Fire TV > Developer Options, once enabled you want to go to about > network to get your ip address.
Now on your chosen OS install and run ADBLink.
Once open, Click New this may take a moment to load.
A window will appear
Enter a description, i called mines FTV 4K, but name it whatever you want.
In the address box, enter your ip address.
Click Save
Under select device, you should see your device already selected.
Click Connect
First time you do this it will probably fail to connect, but if you look at your firestick it is asking for permission, give it permission and then click connect again.
In “Connection Status” it should now say IP connection with the device description on the left hand side.
Next click on ADB Shell
If you enter the follow line of commands, it will tell you what value is currently set to
settings get system screen_off_timeout
Default screensaver value is usually 300000, about 5 minutes.
settings get secure sleep_timeout
For sleep mode, the default value is usually 1200000 which is about 20 minutes.
The max value you can use for your screensaver is 2147460000, you’d think 0 = off but apparently it doesn’t, to change the value you’d enter the following.
settings put system screen_off_timeout 2147460000
Now for sleep mode, just change the value to 0 to turn it off.
settings put secure sleep_timeout 0
if you want, you can use the settings get
commands I provided to check the values were entered, now you can just close and disconnect ADBlink and ofcourse disable debugging.
Hope this helps.