Choosing a RJ45 to WIFI adapater for use with Embedded Server
Your "RJ45 to wifi adapter" could be just cheap WiFi access point.

If you want WiFi only then ESP8266 modules are very popular recently thanks to their price. They can be used as sort of AT-based modem connected to AVR (I've used it like this in weather station, it used roughly 8kB FLASH memory with SHT11, floating point and some logging) or standalone (without external MCU, e.g. running script-based application).

If you'd like to stay with cable you may also want to look at PIC18F67J60 - this is equivalent of ENC28J60 and PIC18 in one chip. Cons: requires pickit2+ programmer, limited FLASH endurance and same as ENC28J60 high current consumption.