Need help with formula to sum certain cells in multiple worksheets depending on a variable

I thought I had this formula right and it seemed to be working fine, but I've hit a snag and can't figure out why it's doing this. Here's the crazy formula I'm working with, I added color coding to try making it a little less confusing:

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Here's what I'm trying to do in layman's terms: I have sheets labeled "Daily_Worksheet" with a number, like Daily_Worksheet0, Daily_Worksheet1, Daily_Worksheet2, etc. and the user can keep adding more sheets. I added the INDIRECT function as a way to get this formula to automatically add the data from each worksheet no matter how many there are. The INDIRECT function is ultimately supposed to end up with the expression "Daily_Worksheet0:Daily_Worksheet#!$A12:$B22" (with "#" being the number of the last sheet). The the SUMIF function is supposed to use that expression to look in all of the "Daily_Worksheet" sheets in cells A12:B22 for the value in cell A4 of the SUMMARY sheet (in this case labor categories like "Chemist"), and if it finds that value it should return the value in the same row from column I.

I'm sure that's all as clear as a bowl of pudding.... If you're with me so far, the problem I'm having is with the part of the formula in italics. The INDIRECT function seems to be set up correctly but the SUMMARY sheet is only adding together the values from sheets Daily_Worksheet0 and Daily_Worksheet4, not Daily_Worksheet0 through Daily_Worksheet4. I can't figure out why though, when I try testing out individual parts of the above formula they work correctly but when I stick them all together they don't. Anyone have any suggestions???

Thanks!!

Well I ended up giving up on that formula above and did what you suggested and I think it'll have to do. Thanks!