If you want to start the sentence with as soon as, you should rephrase it as follows, for example:
As soon as I realized the big box placed outside of my room caused you any inconvenience, I removed it. Now I would like to sincerely say I am sorry.
Realize is used as "I realized I was rude." In your sentence, taking off the parenthetical, you said "I realized the big box placed outside of my room"; it misses a verb. You should remove the comma and which, to obtain "As soon as I realised the big box placed outside of my room had been making you feel inconvenient to go along the corridor during the past 3 days." It is too verbose; "caused you any inconvenience" is a shorter way to say what you are trying to say.
"As soon as" is used as "As soon as she arrives we are ready to go." Your sentence is missing a clause. What did you do when you realized the box caused her/him any inconvenience?
You cannot use a comma before "now I would like to say sincerely sorry" since that is a new sentence.