Use the IRrecord example of this library.
https://github.com/z3t0/Arduino-IRremote
Start decoding some buttons. If I pressed CH+ the result was 20DF00FF with NEC code type.
Here is a tuturial: http://www.righto.com/2009/09/arduino-universal-remote-record-and.html

Once you decode the necessary buttons, arrows and OK button in my case, you can program how to "navigate" the keyboard.
The keyboard starts at "g", it is as simple as 3 times right button and OK for pressing a "k". With this logic you can enter your complete email address and then the password.

The code is really simple, there are several ways to improve it, but I wanted this quick solution for a daily problem and can't wait a long to have it working.
To minimise external components I use the reset button to initialize the sequence.

As you can see, the board is powered with a CR2032 battery. It is necessary to put the clock at 8Mhz to run the Atmega328 at 3V.
I upload a code with this settings, there are probably some extra or unnecessary thing cause I just copy & paste code from an old project. Although, it works! The uC sleeps drawing 8uA.