Windows 7 Laptop Has Wireless Access but No Internet Connection

Question

Dia duit!
Okay, I have been called to the rescue concerning my mother's HP laptop again, and for once I cannot for the life of me resolve the issue.  The machine is running Windows 7 because friends don't let friends run Windows 8.  She has a wireless network
connection, but no internet access when she is in Normal Mode.  However, she does have access in Safe Mode with Networking.  I brought my own laptop over when I came to visit, and it connects to the wireless with no trouble.  My mother has tried
three different web browsers-- Firefox, Chrome, and IE-- so it doesn't seem to be a browser issue.
WHAT I'VE TRIED THUS FAR:
Naturally, given that it worked in Safe Mode, my first thought was a possible malware infection.  In Safe Mode with Networking, I ran Malwarebytes and Combofix.  Sixteen infections were found and removed, but none of them were particularly nasty. 
Afterward I used CCleaner's registry scanner to take care of several registry issues.  I then completely shut down and rebooted the computer back into Normal Mode.  The same issue still persisted.
I then removed and added the wireless network connection under Network and Sharing Center>Manage Wireless Networks.  The laptop still had no internet access.  I toggled the wireless switch off and back on.  No dice.  In desperation
I finally reset the IP and DNS (cmd /c "ipconfig /release" and cmd /c "ipconfig /renew.")  It still didn't fix the problem.
I have never seen this issue when one of the above measures didn't resolve it, so I would be extremely grateful for any advice.
THE IPCONFIG LOG OF DOOM:
Microsoft Windows [Version 6.1.7601]
Copyright (c) 2009 Microsoft Corporation.  All rights reserved.
C:\Users\Donna Hall-McDowell>ipconfig
Windows IP Configuration
Wireless LAN adapter Wireless Network Connection 3:
   Media State . . . . . . . . . . . : Media disconnected
   Connection-specific DNS Suffix  . :
Wireless LAN adapter Wireless Network Connection:
   Connection-specific DNS Suffix  . :
   IPv6 Address. . . . . . . . . . . : fdbd:3ce5:2c36:0:f045:d173:fb1d:3b09
   Temporary IPv6 Address. . . . . . : fdbd:3ce5:2c36:0:98f2:6fc3:f6de:7591
   Link-local IPv6 Address . . . . . : fe80::f045:d173:fb1d:3b09%12
   IPv4 Address. . . . . . . . . . . : 192.168.1.135
   Subnet Mask . . . . . . . . . . . : 255.255.255.0
   Default Gateway . . . . . . . . . : 192.168.1.1
C:\Users\(user name removed)>

Answer 1

Hello,
A few things to try:
1.  Did you check Device Manager to see if the wireless adapter is listed and is enabled?
2.  Did you try to un-install the wireless adapter and have the laptop scan for new hardware?
3.  Is there an icon in the systray for the wireless network connection and if so, what is its state, bars or no bars with the "No" symbol?
4.  Is Windows or another utility controlling wireless connections?
5.  Can you hook it up via a wired connection and check HP's site for updated drivers?