There is every indication that Smith's story of Lehi's transoceanic crossing was supposed to be carefully recorded and that Smith kept track of the time it required to sail from Jerusalem to the promised land but failed to record the number of days because he couldn't remember what had previously been transcribed by Martin Harris.
It's impossible to know what Smith originally recorded in the 116 lost pages pertaining to the voyage. That will ever remain a mystery. But the account we have in the Book of Mormon strongly suggest that Smith originally intended to keep track of the days pertaining to the whole of the voyage although a final tally was never tendered in the record other than,
"for the space of many days". The introduction of going into the ship and being driven forth by the wind begins with the following:
1 Nephi 18 wrote:
8 And it came to pass after we had all gone down into the ship, and had taken with us our provisions and things which had been commanded us, we did put forth into the sea and were driven forth before the wind towards the promised land.
The first marker of the voyage is with reference to the rebellion of Nephi's brethren wherein it seems they began to party and have a good time. Smith failed to record the number of days already had by the voyage. This may be because the event was recorded in the 116 lost pages and Smith couldn't remember how many days he originally gave so he applied the
"for the space of many days", instead of the number of days that actually passed.
1 Nephi 18 wrote:
9 And after we had been driven forth before the wind for the space of many days, behold, my brethren and the sons of Ishmael and also their wives began to make themselves merry, insomuch that they began to dance, and to sing, and to speak with much rudeness, yea, even that they did forget by what power they had been brought thither; yea, they were lifted up unto exceeding rudeness.
The next marker involved Nephi being bound by cords and the compass failing to work. Note that Smith faithfully records the time having passed during the storm as: 3 days.
1 Nephi 18 wrote:
13 Wherefore, they knew not whither they should steer the ship, insomuch that there arose a great storm, yea, a great and terrible tempest, and we were driven back upon the waters for the space of three days; and they began to be frightened exceedingly lest they should be drowned in the sea; nevertheless they did not loose me.
The final marker to prove that Smith was keeping track of the days is when the ship was about to sink due to the storm.
1 Nephi 18 wrote:
14 And on the fourth day, which we had been driven back, the tempest began to be exceedingly sore.
The above time references indicate that Smith was keeping track of how long the voyage took and that the story of the voyage provided both a start and end -- but no number of days! The number of days was left out of the record for obvious reasons. Smith was worried Lucy Harris could have easily scratched out the record and recorded a different number so he had no choice other than to avoid the issue entirely.