Presumably you've increased the number of available IP's to handle the
extra device ? Don't doubt you have but worth mentioning You may
need to manually add the MAC address of the Wi-Fi card in the HP mini
and you may also need to temporarily enable SSID transmission to tell
if the wireless is actually working - unless you have other Wi-Fi
signals around that it would be able to pick up.

Reading up a bit on the HP Mini I think that the Wi-Fi card is a
Broadcom one and these have always been a pain with Linux of any
flavour but usually updating Ubuntu with it plugged into a LAN is
enough to get the driver installed. Check for Hardware Drivers in
System / Administration.

BTW - one point about adding IP addresses to your restricted range on
the router. If you only add one more then this will be used by the
cable so you'd need another one for the Wi-Fi connection ! Maybe it
would be worth increasing the range temporarily until you get the
devices sorted then dropping back to the actual number required,
bearing in mind that wired and Wi-Fi use different MAC addresses ??

I also concur with the comments about security. Hidden SSID isn't a
great bar to anyone getting in these days and unfortunately it's a
fact of encryption that as you increase the strength then the range of
the router drops dramatically !!