Router speed G or N, N300 Router

One thing to keep in mind with Wi-Fi is that the reported and actual speeds are very different.  A 802.11g connection operating at 54 Mbps will usually only get actual speeds of 23 Mbps.  I'll spare you the technical reasons.  If your Internet service is really 60 Mbps, then your HP won't be able to take full advantage of it.  Whether you need a full 60 Mbps to your laptop is a different question, but I did want to point out the difference.

As to getting the RTL8188EE to speak 802.11n, you may have to experiment with changing the advanced properties.  I believe there is a setting called Wireless Mode.  See if there is an option for 802.11n or 802.11b/g/n.  Also, make sure that HT Mode is Enabled.

I also saw some reports suggesting that setting the channel width and bandwidth settings to 20 MHz may result in more stable operation.  But focus on getting 802.11n working before touching either of these two settings.

[Edit: Here is a post on another forum (link) that suggests setting the wireless mode to 802.11b and setting the bandwidth to 20 MHz.  As I mentioned, experiment with various settings.]