The difference between Ich bin krank and Mir ist kalt ist the subject of the sentence.
First, let's have a look at the following sentences:
- Ich bin krank.
(I am sick.)
- Es ist kalt.
(It is cold.)
Here, the subjects are ich and it, respectively. If you want to say that it is me who feels the coldness of the abstract subject it, you can add the dative object mir:
- Es ist mir kalt.
- Mir ist es kalt.
Both sentences might sound a bit strange to a native speaker, but they are correct. In fact, when the verb is a form of sein, then the abstract subject es is often omitted, i.e., it is present only implicitly:
Ich bin kalt.
is also possible but means that I am the subject being cold (my body). Often the expression jemand ist kalt is used as a metaphor to say that someone is coldhearted (see meaning 3. of kalt).
Finally, have a look at
Er scheint mir krank.
(He appears sick to me.)
Here, er is the subject being sick, and mir is the dative object to whom his being sickness appears. The sentence
Mir ist krank.
does not make sense, for it means that there is an abstract it being sick, and I only feel its sickness.