How to change the network type?

First what version of windows are you using?
Vista?

open the network and sharing center, under the name of your network connection it should say public network, click it, and you should be able to alter it to whatever you want.

well i can guess why, on the screen that you cant get to theres a tick box and i imagine theres a similar box when setting up a new network which reads - "treat all future networks as public and dont ask me again"

now, i think i have found an option that might override this

goto the start menu and type in the search box - gpedit.msc
find this branch - computer config>windows settings>security settings>network list
and you should see your network listed on the right, however dont select it, look for the one called 'all networks' select that and select properties, second section down theres a radio button for 'user can change location' select that and click ok.

A wireless router would have saved you a bit of a headache... but at least you got it figured out. Wouldn't a router also have been cheaper? A lot of wireless routers I see are actually cheaper than a good portion of the WAPs I see.