"무슨 말로 이 감사의 마음을 전해야 할지 그저 아득할 뿐입니다"
=> I can't even begin to find the words to express my gratitude for you.
=> I am completely at a loss for words to tell you how grateful I am.
* -ㄹ지 = that/whether/if/how (clause) - forms a noun or adverbial clause.
The -ㄹ ending form represents a possibility or expectation for something to happen, or to occur in due time (i.e. the future), similar to the to-infinitive in English.
Compare 떠날 사람(one to leave), with 떠나는 사람(one who is leaving) and 떠난 사람(on who has left).
-지 at the end makes a clause like "that ...", "if/whether ...", and sometimes "so ... that" (if -ㄴ지). It can be taken as a special noun clause maker (as compared with the general ones like -ㄴ/ㄹ/는 것, -기, and -ㅁ.
As these examples show, it is followed by 알다, 모르다, and other words of perception or emotional reaction.
- 걔가 어떻게 생각할지 모르겠다 = I don't know how he will take it.
- 서울역(에) 어떻게 가는지 아세요? = Do you know how to get to the Seoul Station?
- 어찌해야 할지 아득합니다 = [어찌해야 할지]가 불확실하고 어렵습니다 = = [어찌해야 할지]를 생각하면 두려운 마음입니다 = I feel at a loss (as to) what to do.
- 얼마나 추운지 입이 얼어서 말이 안 나와 = It's so cold my lips are frozen and I can't even talk.
* 아득하다 means that a situation or one's mental state feels very distant and blurry, making it hard to cope with. It can also describe how one is physically light in the head and losing clear sense. In all cases the central meaning is feeling at a loss losing confidence and control.
Here, the author is saying she feels so overwhelmed with gratitude she can't even find the words to thank the people.
Anyway, congratulations on finishing your first professional novel in Korean :-)