This is rather simple. According to Grammar Girl, each other is always two words in English. The AP Stylebook further explains each other and one another, where more confusion occurs.
As quoted from the 2007 AP Stylebook “Two people look at each other. More than two look at one another. Either phrase may be used when the number is indefinite: We help each other. We help one another.”
To sum it up, eachother is not a word in English. Each other and one another depend on the number of people involved.
Clear as mud?