Hi,
According to the [https://downloads.dell.com/manuals/all-products/esuprt_laptop/esuprt_latitude_laptop/latitude-e4310_service%20manual_en-us.pdf|service manual] in ''BIOS > System Configuration'' (p.3) there should be setting options for ''keyboard Illumination''.
Have you tried updating to the latest [https://www.dell.com/support/home/au/en/aubsd1/product-support/product/latitude-e4310/drivers|BIOS version] (if it is not already at the latest version) and check if the option is now available? (select BIOS in Category box)