If the AirPlay mirror icon isn't displaying, your iOS device isn't able to see the computer for one reason or another.

If you're on a home network, the firewall on your computer may be preventing the devices from communicating. It's also possible that the firewall on your router is preventing communication.

Microsoft has included instructions for allowing applications through the Windows firewall here:

If you're using a Mac, Apple has provided similar instructions here:

Corporate networks often block a number of ports from communicating across networks. You may need to have your IT department allow TCP ports 7000, 47000, 7100, 49228, 50259 and UDP 62572, 54780 to be open to communicate. The network must be also allowed to run Bonjour and mDNS in Multicast mode. Unless special steps have been taken, Reflector will likely not work across subnets or VLANs.

Apple has provided more instructions on this issue here: http://support.apple.com/kb/TS4215