When replacing a PS, how do I know I'm safe if device doesn't list current?
You can easily calculate it starting from the power. Since \$P=V\cdot I\$, then \$I=\frac{P}{V}\$. In your case \$P_{max}=5W\$, while \$V=12V\$, then \$I_{max}=\frac{P_{max}}{V}=\frac{1.5W}{12V} = 0.125A = 125mA\$. Since your PSU can provide up to 2.2A you are safe. Keep in mind that your replacement is way too big, if you have space or possibly heat constraints you might want to buy a smaller PSU.

If you want to better understand how to choose a power supply have a read at this question.