You cannot set the timeout to "never" on your system.
The feature is (at this writing) available on business class machines that include the BIOS > Advanced section.
Yes, I know this is not the answer you wanted and I wish it were otherwise.
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.
Section: The backlight stays on for only a few seconds
If there is not a setting for the backlight timing in the BIOS, this behavior might not be supported on the notebook. Check your notebook model's product support documentation to confirm support.
In general, and there might be exceptions,
Business class computers offer the BIOS Advanced section.
Examples of business class systems include and are not limited to: Zbook Mobile Workstations, ProBook, EliteBook, EliteDesk, and others.
The backlit keyboard timeout option is included in the Advanced section as appropriate (assuming system has a backlit keyboard).
At this writing, consumer class systems may not include the Advanced BIOS section.
Examples of consumer grade systems include and are not limited to: Spectre, Pavilion, Envy, Notebook.
Device Home Page - References and Resources –Learn about your Device - Solve Problems
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, Manuals / User Guides, Service and Maintenance Guide (Replacement Parts and Procedures), Product Information (Specifications), more
When the website support page opens, Select (as available) a Category > Topic > Subtopic
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