Breakdown of Í janúar er ég að lesa bókina til að auka þekkinguna.
vera
to be
bókin
the book
ég
I
að
to (infinitive marker)
Questions & Answers about Í janúar er ég að lesa bókina til að auka þekkinguna.
The lesson is about til þess að, but this sentence just uses til að. Is there a difference?
No, they mean exactly the same thing. In everyday Icelandic, people very often drop the þess to make it shorter. You just use til að followed directly by the infinitive verb, which here is auka (to expand or increase).
Why is the word order er ég instead of ég er?
This is because of the verb-second rule. Whenever a sentence starts with something other than the subject, like the time phrase Í janúar, the verb er must take the second slot. This pushes the subject ég right after it.
The English translation says my knowledge, but I do not see a word for my in the Icelandic sentence.
That is a great observation. Instead of using a possessive word for my, Icelandic usually just tacks on the definite article when the owner is obvious from context. Here, þekkinguna literally means the knowledge, but it is naturally understood as your own knowledge because you are the one doing the reading.
How does the continuous am reading part work when it is split up by ég?
The ongoing action is formed with the verb to be (er) plus að plus the main verb (lesa). Because of the verb-second rule, only the conjugated verb er moves up to the second position. The rest of the phrase (að lesa) stays later in the sentence, wrapping around the subject.