lawcrisis,
Welcome to the HP Forum.
Are you sure you are using WEP on that router? Unusual - the protocol is outdated and more vulnerable than what is considered the more common standard in today's home network.
It is more common that you would be using WPA.
The connection type must match on the router and the computer, of course.
Reference:
HTG Explains: The Difference Between WEP, WPA, and WPA2 Wireless Encryption (and Why It Matters)
You might take another look at the Troubleshooting tips herein:
Troubleshooting Your Wireless Network and Internet Connection (Windows 8)
You can manually create a "new" connection and bypass the existing connection that is not working as expected:
HP PCs - Creating a Wireless Home Network (Windows8)
- You might consider using a Static IP on the computer.
I admit I might be missing the point.
Commentary
You might also check your Security Software -- you should not be seeing a "Public" network setting on a home network (in my opinion). Public Network connections tend to be (at the least) restricted and annoying -- at the most, you may not be able to retain the connection needed. Public / Home-Private / Work Connection type is something your security software usually controls and not something the computer controls in and of itself.
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Although I strive to reflect HP's best practices, I do not work for HP.
