I’m glad she is recovering from her mishap. Keep in mind that at 11 weeks old, she is still a baby. Even more so than human babies, puppies explore their world by putting everything into their mouth. For some dogs, this also includes swallowing it, which can cause a health risk as you discovered. Coming down hard on a puppy, though, will only make the problem worse and damage your relationship with her. I would take a 4-prong approach:
1) Rule out any reasons why she might be excessively interested in swallowing things. A friend had a golden who constantly grabbed and swallowed everything in sight. After lots of failed attempts to fix the problem (and multiple trips to the vet), a vet finally told her that she was not feeding her growing puppy enough. Sure enough, she upped the amount of food she was giving her and the unwanted behavior dropped dramatically.
2) Teach her that giving you things is a good idea. Often, dogs start swallowing things because they have learned that we take “good” things from them. Do lots of “trades” in a training environment. Start with easy stuff, and then work up to harder things for her. There are lots of videos about this under the category of “resource guarding.”
3) Since she is doing well with “Leave It” training when you obviously drop the item, you can now take it to the next level. Go to wherever you are going to be training WITHOUT the pup and place a tempting object or treat in the area. Now bring the puppy out and use this for your Leave It training. Don’t use corrections, but make sure that she cannot reach the item (stop so that the object / treat is just beyond leash range). If she is interested and pulling to get to the item, wait her out. Just stand there until she turns to look at you with a “What’s going on?” look. IMMEDIATELY praise and reward, which should result in her running back to you to get her reward. Repeat a couple of times and then continue with other training (leave the object / treat on the ground, but move away from it for your other training). After a few minutes, repeat the “Leave It” exercise. The first few times I wouldn’t even use the “Leave It” command, since there will likely be a long pause before she gives up trying to get to the object. Once you see that she is starting to catch on, then you can introduce “Leave It” and reward for response.
Once she is really catching on, sprinkle a few objects around the area before bringing her out. Then, start doing the same exercise with tempting things that just happen to be on your walk. By this time, she should be starting to generalize what “Leave It” means, so you can start using the cue on your walks. It is crucial that she NEVER is able to get something once you have told her “Leave It,” so you will need to decide when you want to use the cue. Do NOT say “Leave It” if she already has the item or you can’t stop her from getting it. It is too late, and better to not use the cue at all than to have her learn that it can be broken.
4) When you are not able to actively train as above, it is time to manage the situation. Make sure all forbidden objects in the house are out of her reach. Puppy-proof your home and keep it that way. When you go on walks, you will need to be hyper-vigilant for a while. Try to spot any enticing object before she does and either avoid it or use it for “Leave It” training.
If she grabs something before you can stop her, don’t scold or chase her. Make a quick assessment. If the object doesn’t pose an immediate threat to her, just ignore it. Correcting her or trying to take it from her will just make it more appealing to her. I have often found that this “ignore it” approach quickly results in the dog picking up fewer things because I have taken the fun out of the game. If necessary to get something away from her, one possible approach is to pick up a different item and pretend that it is the most exciting thing around. Tease her with it a bit, and then let her “trade” you for it. You can even do this trade back and forth a few times, ending with a huge jackpot. If she enjoys tug, you might use a tug toy that you reserve just for this purpose. Praise her as soon as she gives you her prize, and then play with the new object with her briefly before calmly ending the game and proceeding with your walk.
If you must remove something from her mouth because of the hazard, do it calmly and matter-of-factly without giving any cue such as “Drop.” (Work on training “Give” or “Drop” separately so that eventually you will be able to use these commands, but trying to use them now will just backfire on you.) You want to make it as much of a non-event as possible. Needless to say, the pup must be on leash to do any of this since you can’t control her when she is off leash and the game of chase to catch her becomes an incredibly high value reward for her to grab undesirable objects.
Good luck! Some of this behavior will subside with maturity as long as you don’t make it rewarding for her. Remember that negative attention is still attention and can be highly reinforcing. Letting the situation turn into keep-away make it even more rewarding. It can be incredibly frustrating and even scary when you have already had to make an emergency trip to the vet, but remember that it is a normal part of puppyhood.