Just an update: The air filter/primer assembly arrived today:

http://tinyurl.com/o3rlm7q

Took 5 minutes to install it and after 2 primes, the mower started right
up. Problem solved.

Thanks again for all the help and advice. This still wasn't that
expensive of a repair and I know it would have been much more if I took
it to the local lawn mower repair shop.

Jason

My Toro 20045 started misbehaving late last summer getting harder and
harder to start. After googling and watching Youtube, suggestions seemed
to be to clean and/or replace carburetor. Following that advice, I
carefully removed the carburetor, unscrewed the bowl and cleaned the jet
holes of the bowl jet and then proceeded to clean all of the indicated
holes of the carb. I did the cleaning first with shots of carb cleaner
spray followed by trash can tie wires through the holes and finally used
compressed air on each hole. I also checked the action of the float and
value and it seemed to be moving normally. Before replacing the carb, I
replaced the gaskets on each side. Once reinstalled and with several
primer pushes, still no start. My mower does start first pull and keeps
running only IF I remove the air filter cover, take out the air filter
and spray carb cleaner or gas into the carb entrance. Then I can
actually mow the grass once the filter is reinstalled but once it runs
out of gas, I have to go through the process of removing the filter
again to give it a shot of cleaner.

Last summer, the primer bulb had cracked to the point where it wouldn't
press or release properly anymore and I replaced it. This is the type
of primer/ air filter assembly the mower has:
http://www.amazon.com/GENUINE-OEM-BR...7FXY6FY435JQ40

I notice that when I have the air filter removed and press the primer, I
don't see anything happening in the carb chamber. Shouldn't I be seeing
squirts of fuel entering the carb when the primer is pushed? I've
suspected that the no start problem may lie within this primer/ air
filter assembly, but I am not sure how to test it. I sent compressed
air into the short hose as seen in the image of the amazon link. What
is the short hose for? Is it supposed to pull gas from the tank when
the primer is pressed? When I remove this hose along with the air
filter assembly, should I be seeing gas dripping from that areas of the
mower. Obviously I do when removing the carb hose.

At this point, I think the problem lies in either the carb itself or the
air filter/ primer assembly. Before I run out and buy one or both
items, is there any way I can definitively test each item first? I wish
I knew more about how the primer assembly is supposed to work also.

I spent the weekend trying to get the mower starting the way it is
supposed to, but to no avail. The good news is that I got the yard
mowed, but each time I would have to refill the tank with gas, I'd have
to remove the air filter, spray a bit of carb cleaner into the carb, and
start first pull. This won't be fun if I have to do it all summer long.

Any help would surely be appreciated.

Thanks in advance,
Jason