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. Thanks.
Man of Honour
Hi and welcome to the forums. 
A quick Google found this supposed compatibility list for your laptop which includes several cpu's. You may need to flash the bios to support a later cpu. There are only two choices on Asus website, versions 306 and 308 so grab the latest one. You can find them here. You also need to be aware that any more powerful cpu's are going to put out more heat which can be a problem with laptops and their limited cooling. Apart from that there is actually very little information available on this laptop.
A quick Google found this supposed compatibility list for your laptop which includes several cpu's. You may need to flash the bios to support a later cpu. There are only two choices on Asus website, versions 306 and 308 so grab the latest one. You can find them here. You also need to be aware that any more powerful cpu's are going to put out more heat which can be a problem with laptops and their limited cooling. Apart from that there is actually very little information available on this laptop.
Don
From what I can see the F5SL uses a SiS 671DX + 968 Chipset.
In theory that chipset supports up to a 1066Mhz FSB and supports penryn based cpus. However the BIOS may be a limiting factor as to what is actually supported/recognised.
Cost effective chips worth a try would be T7400 (2.17Ghz, 667FSB, 4Mb Cache), T9400 (2.53Ghz, 1066FSB, 6Mb Cache), P8400 (2.27Ghz, 1066FSB, 3Mb Cache), all of which can be found used for £5-£10. The more powerful models e.g. T7800/T9500 are noticeably more expensive ~£20-£30 so probably not worth the risk:
references:
https://en.wikipedia.org/wiki/List_of_Intel_Core_2_microprocessors#Core_2_Duo_2
http://forum.notebookreview.com/threads/asus-notebook-upgrade-please-help.570598/
http://www.computer-specifications.com/specifications/ASUS-F5SLNotebook-Specs.html
http://www.cpu-upgrade.com/mb-SiS_(chipsets)/671DX.html
In theory that chipset supports up to a 1066Mhz FSB and supports penryn based cpus. However the BIOS may be a limiting factor as to what is actually supported/recognised.
Cost effective chips worth a try would be T7400 (2.17Ghz, 667FSB, 4Mb Cache), T9400 (2.53Ghz, 1066FSB, 6Mb Cache), P8400 (2.27Ghz, 1066FSB, 3Mb Cache), all of which can be found used for £5-£10. The more powerful models e.g. T7800/T9500 are noticeably more expensive ~£20-£30 so probably not worth the risk:
references:
https://en.wikipedia.org/wiki/List_of_Intel_Core_2_microprocessors#Core_2_Duo_2
http://forum.notebookreview.com/threads/asus-notebook-upgrade-please-help.570598/
http://www.computer-specifications.com/specifications/ASUS-F5SLNotebook-Specs.html
http://www.cpu-upgrade.com/mb-SiS_(chipsets)/671DX.html