Unfortunately, at this writing, on your type of computer (consumer class), the timeout on the backlit keyboard cannot be set to "never" (leave on all the time).
BIOS – Backlit Keyboard and other Advanced Options
This is one of those situations where the how-to document mentions that a particular feature might not be supported on every computer. The document does not offer an exact explanation because the document is not written for YOUR computer.
Reference and Resources – Places to find help, solve problems, and learn about your Device
NOTE: Content depends on device type and Operating System
Categories: Alerts, Warranty Check, HP Drivers / Software and BIOS, How-to Videos, Bulletins/Notices, How-to Documents, Troubleshooting, User Guides / Manuals, Service and Maintenance Guide (Replacement Parts and Procedures), Product Information (Specifications), more
When the website support page opens, Select (as available) aCategory>Topic >Subtopic