This Answer is obsolete. The operation of mirrordirector has changed.
Using terminal or your favorite text editor you can define a specific mirror to use in
Just comment out the line
deb http://mirrordirector.raspbian.org/raspbian wheezy main firmware
So it looks like this
#deb http://mirrordirector.raspbian.org/raspbian wheezy main firmware
And replace with a mirror close to you from list below
The list of mirrors can be found here
I (being in the US) use the lease Web mirror
Problem seems dns related (except for slow ping when you used actual ip of Google.
Update your /etc/network/interfaces eth0 line to include (or update existing) to be (as last line of section)
dns-nameservers 184.108.40.206 220.127.116.11
Those are Google name servers. Just to see if get better performance. If you would like less prying eyes than Google, there are other free providers out there.
As far as slow ping when pinged ip address of Google, I would reboot router and/or modem.
If adding to /etc/network/interfaces didn't work add the following as last line of /etc/resolv.conf
nameserver 18.104.22.168 22.214.171.124
And then restart or just restart networking
Also if you could update answer with output of
You do this via terminal. It will show us the local address of the Pi. Also, please update answer with local ip of computer that is working, or the local address of the router. For example mine is