Not only should verses 18 and 19 be read together, but they should also be read in contrast to verse 17's redemption of the repentant. The fall of the great and abominable church comes precisely because it is not able to repent. The fall of the kingdom of the devil comes on the heels of the fall of the great and abominable church because they are poetic equivalents. Just as Isaiah used poetic repetition, so does Nephi repeat the same concept in different terms. The great and abominable church and the kingdom of the devil both refer to the devil's reign - but highlight different aspects of it.
The great and abominable church operates in a temporal sphere. Thus the great and abominable would be the devil's organization on the earth. Just as the kingdom of God pertains to the eternal realm, so too does the kingdom of the devil point to the eternities rather than the time delimited earthly domain. The devil will fall, and will fall both on earth and in the heavens. He will have no dominion at all - and will be completely subject to God.