The following answer is wrong. Please see the comments to know why.
Let us label the first person to speak as A, second person to speak as B and so on.
Let's first look at the general case. Number of hats that A can possibly see :
6,7, 8 and 9.
Case 1: A sees 6 or 9 hats
If A sees 6 hats then he will know on his first turn itself that he is wearing 2 hats. This is because it is given that he can only wear 1 or 2 hats. It is also given that total number of hats is 8,9 or 10. 6+1 = 7 which is not possible. The only possibility when he sees 6 hats, therefore, is that he is wearing 2 hats.
Similarly, if he is seeing 9 hats then it has to be the case that he is wearing 1 hat.
So, if A calls out 1 or 2 on his first turn, then B,C,D and E will know that they are wearing a total of 9 or 6 hats respectively among themselves. They will then be able to figure out their hat count on their first turn itself.
Case 2: A sees 7 or 8 hats
If A says, "I don't know" on his first turn then it will indicate to B,C,D and E that there are a total of either 7 or 8 hats among them 4.
Case 2a : There are actually 7 hats among BCDE
In this case, everybody among B,C,D and E who is seeing 5 hats as the total hat count among the other 3, will be able to guess that he is wearing 2 hats. This is simply because they know that among B,C,D and E, there are a total of 7 or 8 hats because A said, "I don't know" on the first turn. Now, when they see 5 hats among the other 3, it becomes obvious to them that they are wearing 2 hats and they call it out on their first turn itself. This, 3 of these 4 people will be able to guess their hat count on their very first turn.
Case 2b: There are actually 8 hats among B,C,D and E. In this case, in the first round, everybody would see 6 hats and they would say, "I don't know." This is simply because, during the first round, when B,C D and E would see 6 hats total on the other 3 people then they would not be sure if they themselves have 1 or 2 hats on their hats. This is because both B+C+D+E = 7 and B+C+D+E= 8 are possible.
Now, very importantly notice that 3 different things happen in cases 1, 2a and 2b .
In case 1, everybody is able to guess their hat count in the first round.
In case 2a, even though A says, "I don't know", in the first round, people among B,C,D and E who are wearing 2 hats each are able to guess their hat count in the first round itself.
In case 2b, everybody says, "I don't know" in the first round.
So, when A says, "I don't know" in the first round and yet, some people are able to guess their hat count then it is indicative that the only possible case is case 2a i.e there are 7 hats among B,C,D and E. Thus the only person among B,C,D,E who is wearing 1 hat will be able to guess his hat count after he hears anybody among B,C,D, E guess their hat count. A, however, will never know his hat count.
Also, when everybody says,"I don't know" in the first round then it will tell everybody that they are in case 2b and that there are 8 hats among B,C, D and E. In the next round, B,C, D and E will be able to guess their hat count. A again won't be able to tell his hat count.
So, the answer is that in the case of 9 total hats, there are either 7 or 8 hats among B,C,D and E. This means that the group is in case 2a or case 2b. And we have proven above that, in either of these cases, everybody except for A will be able to guess their hat counts.