How can I be more active during the day with a desk job?!

A couple of things that I've been able to do. They may not be feasible for your work but I'll share anyway.

1. Bluetooth earpiece: I ordered a bluetooth earpiece for my work phone, now whenever I'm on a phone call I make a point of being on my feet and moving while I speak
2. Walking meetings. I have a rule. If There are 3 or less people in the meeting and no one has material to present then it's going to be a walking meeting. Weather permitting I'll meet people and we'll head out of the office and talk while we walk. It's a great way to add some movement and gets you outside in the fresh air.

Pretty much any incidental movement is gonna be of benefit. Even you are tethered to your desk doesn't mean you need to be anchored to your chair. set a 50 minute timer. When it goes off make sure you give yourself an excuse to get up for whatever reason (talk to someone instead of emailing, picking up a printout, getting some water, anything). At the very least you can stand up and stretch a little to get the blood flowing.

I also walk during my lunch break, and I'll occasionally walk to the train station a few stops away in the afternoon if I've had a particularly inactive day.

I am a teacher so spend a lot of time standing while actively teaching, but I am also the head of department so have a mountain of admin to do. I have standing desks in my office and in my classroom. I only sit down to eat lunch, the rest of the time I am on my feet all day. It took a bit of getting used to, but now it is great - I barely notice it at all. It doesn't really get me fit, but I no longer have a sore back or neck and when I do exercise (mostly run or walk) my legs don't feel it the same way they used to.

I'm just seeking some tips and suggestions for getting in more activity during the work day. I tend to sit at my desk constantly and have a hard time prying myself away from work. It seems the deadlines are always "NOW!"
Don't accept unreasonable deadlines or that becomes the default expectation. "Lack of planning on your part doesn't constitute an emergency on my part".
Manage your workload and your manager or both will drive you instead.
Build exercise into your commute.
Eat lunch at your desk and use your lunch break for exercise.
Use the stairs.
Go and talk to people rather than send emails to people nearby.

Any tips on how to set some work boundaries so I can add more activity in my day? It's gotten to the point where my ankles are a little swollen/tight from not getting enough movement in throughout the day.
Hope you have recorded that your work is impacting your health with your manager and HR?
My boundaries were setting a reasonable professional working day in terms of both time and workload. That was a lesson learned from my early 40's when I had to walk away from a 24 year career due an intolerable workload which was making me ill, depressed and with a major drink problem.
If you don't set boundaries then you will find "if you want something done then give it to the busiest person in the office" is very true.
Don't let technology blur the boundary between work and personal time - unless you are paid to be on call don't be.

Has anyone had successes with losing weight with a demanding desk job?
Yes - I had a full time demanding office job for my client, long commute, also running my own business - remember none of that makes you eat too much. Make sure your eating is to your needs and not stress/emotional/tiredness eating.
I also managed an extensive exercise routine (some during lunch breaks, some late evening, a lot at weekends. But remember it's the not eating too much that is the primary driver for weight loss.