TranslateIt class using the Yandex translation service
Some ideas:


If the arguments are a fixed list, don't use **kwargs. I find **kwargs is only really useful if you have some middleware code which needs to forward an arbitrary set of arguments to another piece of code. That way people don't need to rely on your documentation (or when the documentation inevitably gets out of date, the code) to understand which parameters the method takes. Using named parameters you can also get rid of loops over **kwargs.
IMO attributes and constants shouldn't be documented at all. Any IDE will be able to auto-complete them, and their names should be sufficiently self-explanatory that you don't need extra documentation.
Starting a method name with check implies an assertion is happening. Use get when that is what you're doing.
This is a personal style item, but I prefer that the interesting/important code is at the top of the file. _checkEncoding is not that interesting.
If key is a secret it should not be hard-coded.
Don't second guess TranslateIt's API. For example, don't verify the max size on the client side. Instead you should rely on the API to tell you when the size is too large and forward the information to the user. This way your implementation is simpler and more importantly doesn't get out of sync with the API.
You should use the with open() pattern in _checkEncoding.
Use pep8 to fix style issues like method names. 
Your script is not usable from a command line.
Why make yandex_string a field? It is only used once, and is never changed, so you might as well treat it as a constant.
The code is not testable as it stands. This is too big a subject to expand much on here, but suffice it to say that translateIt does much more than one thing.