I have the exact problem in 13.10 as this user
Why are there two Wifi indicators in the panel?.

Here are some screenshots:

Here are some screenshots from another user: http://ubuntuforums.org/showthread.php?t=2183020&p=12825563

ifconfig and iwconfig outputs

$ ifconfig

lo        Link encap:Local Loopback  

          inet addr:XXXXXX  Mask:XXXXXXX

          inet6 addr: ::1/128 Scope:Host

          UP LOOPBACK RUNNING  MTU:65536  Metric:1

          RX packets:2243 errors:0 dropped:0 overruns:0 frame:0

          TX packets:2243 errors:0 dropped:0 overruns:0 carrier:0

          collisions:0 txqueuelen:0 

          RX bytes:209889 (209.8 KB)  TX bytes:209889 (209.8 KB)



wlan0     Link encap:Ethernet  HWaddr XXXXXXXXX  

          inet addr:XXXXXX  Bcast:XXXXXXXX  Mask:XXXXXXX

          inet6 addr: XXXXXXX Scope:Link

          UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1

          RX packets:5925 errors:0 dropped:0 overruns:0 frame:0

          TX packets:3361 errors:0 dropped:0 overruns:0 carrier:0

          collisions:0 txqueuelen:1000 

          RX bytes:2951818 (2.9 MB)  TX bytes:630579 (630.5 KB)



$ iwconfig

lo        no wireless extensions.



wlan0     IEEE 802.11abgn  ESSID:"XXXXX"  

          Mode:Managed  Frequency:2.437 GHz  Access Point: XXXXXXXX

          Bit Rate=72.2 Mb/s   Tx-Power=15 dBm   

          Retry  long limit:7   RTS thr:off   Fragment thr:off

          Power Management:on

          Link Quality=49/70  Signal level=-61 dBm  

          Rx invalid nwid:0  Rx invalid crypt:0  Rx invalid frag:0

          Tx excessive retries:153  Invalid misc:472   Missed beacon:0