I know I'm not Japanese, but seeing as 離 (in the case of 離れる) means "to be separated" or "to leave" and as 苦 (in the case of 苦痛) means agony, I'd have to say you have the right combination. However, using goo's dictionary, I got the definition 「心身の苦しみを離れること」which means "to separate (oneself) from mental and physical suffering." Basically, it's a word used mostly in Buddhist contexts. This might not be exactly what you were looking for. The English definition you provided led me to believe it meant something along the lines of "agony caused by being separated from something/someone", so I assume the same could have possibly happened to you.

Actually, you are free to use any combination of り and く that you like. There are probably hundreds of them. There are special sites dedicated to giving Japaneses names to English names. Google one of them.