He meant, on the fifth day from this day, I will arrive at dawn, from the east.
Note that this does not imply that the battle of the Hornburg last 5 days. Aragorn, Theoden, Gandalf and the rest of the party had not reached the Hornburg yet. As I recall (I have not have "The Two Towers" on hand) they were still a day's journey from reaching the Hornburg when Gandalf made that announcement and left. Then after they arrived at the Hornburg there was another day and a night during which Saruman's army arrived and besieged the fortress. Then the battle began and lasted a day and a night.
The Tale of Years, Appendix B, says that Gandalf healed Theoden at Edoras on March 2. On March 3, Theoden had taken over the defense of Helm's Deep. The battle of the Hornburg began on the same day, and on March 4 it was over and Gandalf and Theoden set out for Isengard. There is a discrepancy here which cannot be resolved, because 5 days did not elapse between Gandalf leaving to gather Erkenbrand and his scattered forces and his arrival on the "fifth" day..
Note also, that Aragorn trusted Gandalf to arrive on time, and convinced Theoden to charge from the Hornburg with the knights of Rohan on the morning of the fifth day. The Ents and the Huorns had arrived during that night, and Gandalf and the men of Rohan led by Erkenbrand charged from the east at the same time as Theoden's knights charged, resulting in the complete destruction of Saruman's army.