How long do I need to drill with my current unit and MOS before I can switch to 11b?

Ahh... Adulting.... Where you have to honor the commitments you've agreed to.
As they've pointed out, your commander is not required to allow you to switch, but you are required to honor your commitment. There is no set time (that I'm aware of) and it's usually driven by local policy.
There are a few ways, all at the commander's discretion. You can move and request a transfer, request to transfer to an Infantry unit, or request to be released to the Active Duty. You're only going to work at your unit 2 days a month, so it's hard to imagine that your lifestyle will be significantly altered by moving to 11B for those 2 days a month. If you don't feel challenged where you are at, I suggest going active and joining the Airborne Infantry and trying out for RASP to become an Airborne Ranger.
Shoot for the moon and if you miss, you'll hit the stars.

In 1974 my CO said it best in the 82nd ABN, "No matter what you do in the Army, Remember, your primary job is infantry". And, don't tell me if you are a cook, supply or you repair M-16's in the arms room, if the enemy is about to overrun your position, "You Become Infantry", and pick up a rifle and fight for your life.

If you’re in the NG need to find an infantry unit that will take you, give you a paragraph line number as as payroll number let both Readiness NCOs know what your plans are and do a 4187 requesting to reclass to 11B. I think you have to be at your current unit for a year.

Depends on a lot of things. Are you prior service or did you enlist in the Guard? How long have you been at your present position? How much of your enlistment is left?

In General, if you enlisted in the Guard and they sent you to a specific MOS, they probably won't want to change it unless the MOS you are in is over strength and the one you want to go to is under strength. You should be able to reclass at re-enlistment as long as there are slots available in the State for your rank. That should be a problem because you can serve in any Non NCO slot in an Infantry Co as an E4 and I would be shocked if an Infantry unit was overstrength.

Just as a word of caution. The operational tempo of Guard Combat Arms units can vary a lot between units. If you are a BCT assigned to a unit like the 10th, you probably have access to more schools and a harder training schedule than if your were with the 35th Div. Army National Guard. I support the 35th as a Combat Engineer and don't means this to be derogatory and they are a good unit, but it's clear that certain units are expected (and funded) to maintain a better state of readiness than others.

It all depends on your unit and CoC. You may have a set time of 2 years that you have to fulfill in your current MOS (depending on how much training you received/money the government put into you). I've knew guys who have re-classed in their first 6 year contract. If you can get conditional releases from your Battalion you'd have a shot. 11B reclass is only 2 weeks. If you just got to your unit out of AIT then you'll probably have to bite the bullet for awhile. Showing up to a new unit and asking to reclass first thing doesn't look good.

If you are in the National Guard, it is generally a year. You need to find an infantry unit and slot that you would like to transfer into and get the contact information for their readiness NCO. Pass that information on to your current readiness NCO, so he/she can make contact and get the para/line for the position and initiate the DMAIL45 for transfer. Your chain of command up to the BN CO has to sign off as well as the receiving unit. They don't have to allow you to transfer, but I have rarely ran into a unit or commander in the NG that will force a soldier to stay in a unit or slot that they hate... unless there is an upcoming national exercise or deployment. Good luck!

Unless you’re doing it because of a hardship of some sort, you may have to ride out your current enlistment and then re-enlist and reclass.

I usually you can switch after 6 month.
You have to wait 6 months to switch how ever that if you command allows you to switch if they don't let you then you have to wait till you are up for reenlistment and then you can switch when you drop your reenlistment packet or you get out and reenlist with a recruiter and then usually they let you switch then onless they really need you back in your mos.
11B is hard to get out of because they are always needing infantry.

It seems like you got sold a bill of goods from your recruiter. Why didn't you contract for 11X in the first place? Can I assume that your are in the Army Reserve rather than the National Guard where the combat arms MOS's are TOE? Another question.... why didn't you go "active duty" 11X in the first place, most recruits who want Infantry want an AD contract and the sign up bonuses are considerable.

You have to convince your Unit to transfer. Provide them with a letter explaining where you’d like to transfer to any why.

A 91F has more practical experience in the Civilian World than 11B. It also helps to make contact with the Unit you’d like to translated to. Maybe even make a site visit and talk with their Readiness NCO about the Unit. Regardless, type a letter and send it up your Chain to your Commander and bother them until you’re transferred. Hopefully you’re not flagged.