Depending on your start point I'd go with Damen overall. Once you get past Chicago southbound it's fairly open. Just use lights and take the sidewalk through the viaducts.
Could you share your starting point? The North Side is a pretty broad concept. In general, I agree with Fran that Damen is a good north-south route. If I were coming in on Damen, I'd hit Blue Island from Damen to 26th and Western, then take the sidewalk, which is lightly used, the two blocks to your work. I know, shame on me for recommending the sidewalk, but I would not be biking Western Ave. in that area. I take Archer all the time, but I wouldn't consider it one of the more most bike-friendly streets available.
Looking on Google Maps myself, and starting at Lawrence and Damen and going to 2801 S. Western via the bike setting (not the default), it seems to me that going to the LFP would take you way too far east and then make you go too far west on the other end. I've ridden from your area past where you're going, and I believe that Damen is your best shot, to Blue Island and then to 26th and Western. From what I see, trying to take the LFP is going to add a good five miles to your trip. Am I missing something? Maybe it's because you're starting your Google Map trip from downtown rather than your actual starting point. Even from downtown, if you took Archer very far you'd start going south further than you want. Archer to Cermak to Blue Island could work, but I wouldn't say those are the easiest streets to bike when traffic is bad, especially Archer. Maybe you should try a few different routes and see which ones you like the best. None are completely ideal, which is par for the course in the big city, I suppose. 18th or 21st to Blue Island could work, but they you lose out on some of the slantiness of Blue Island.
If you travel to and from the LFP, you're still using city streets. Have you tried Central Park/Monticello? When you pass Waveland, go thru the park to Addison, then under the Kennedy to Lawndale, south and west back to Central Park. The biggest problem is at Division. The stop lights at Division & Grand are a pain - they're so long. Monticello is probably a faster way across. Then for some damn reason Central Park is one way north between Division & Chicago. You can either go thru or go around. South to Douglas Blvd, East to Sacramento Dr, South on Marshall Blvd, South on California Blvd, East on 31st Blvd. And all those streets after Central Park have bike lanes. Then it's just a couple blocks north on Western to your destination.
If you take the LFP, try 18th St to Blue Island.
I echo the sentiment that Damen makes a lot of sense for this route.
The United Center section of Damen can be a very crappy few blocks though as Damen turns into a four lane street with no shoulder or parking lane. But in the context of your entire trip, this is a pretty small problem.
As for your originally proposed route, I can't discourage riding on Archer enough. Southwest of Throop it has no bike lanes and motor vehicle traffic can move very fast and there are numerous sites where the outside traffic lane gets squeezed pretty tightly. (usually at intersections to allow for a left turn lane).
If you insist on trying it. One can take the lakefront path to just south of Soldier Field where one can cross the Metra tracks via an overpass dumping you onto South Calumet, just south of 18th. Take 18th west to State, take that south for a block or so to the northeastern terminus of Archer.
I agree with Damen being a better route, but the OP wants to take the LFP, so with that in mind I'd agree with the advice to take Blue Island - I'd probably get off the LFP at 18th and take that west to Damen.
If you insist on taking the LFP, 18th is chill compared to Archer, though you still have to watch out for shitty drivers in certain specific spots, such as the Dan Ryan intersection.
I am just curious if there is some reason you are wedded to taking the LFP, given how out of the way it is from your starting and ending points.
I get that many people feel more comfortable on the LFP, but you'll be riding just as many miles on city streets to get to the LFP and back as you'd be riding if you took a more direct route.
Maybe you want the exercise/conditioning of the uninterrupted LFP riding?
Just curious more than anything--not trying to be critical. :)
I cannot speak for the OP, but my own (long) commute gives me a technically-shorter-distance (maybe 21 miles instead of 25) route home using MLK Drive, but I choose the LFP every time because it is so, so much less stressful. No one in a car is trying to kill me and I can ride nonstop from Marquette to the Loop or Diversey or wherever. I don't blame the OP for choosing the LFP, whatever their reasons.