At what point does the Lady get "called in"?

Yeah, agreed. Sigil's been engulfed in war before without the Lady of Pain intervening. There's no indication that she interfered with the Mercykiller War (300 years ago, when all the factions tried to kill each other and three factions went extinct after the Mercykiller factol was assassinated) or the Doomguard War (200 years ago, when the Harmonium and Doomguard battled over who would become Sigil's police force). She didn't interfere with the recent Faction War either until after it had ended (when she banished the factions from the city or demanded they dissolve their leadership).

There's even been invasions of Blood War armies that lasted for months without the Lady of Pain doing anything (this is how the Slags came to be).

The thing about the Lady of Pain is, you can't predict her and she won't come on demand. There's nothing you can do that will guarantee she'll show up. It has to be something that, in her estimation, won't go away on its own (or with the help of Sigil's mortal police). It has to be a threat to the whole city. She doesn't care if large numbers of people are murdered, or starving, or enslaved. It's only when things are so bad that people begin to doubt her power that she acts.

As far as most Sigilians are concerned, the Lady of Pain might not even exist (but few are willing to risk saying so out loud, and the Lady likes it that way; if they did start saying that out loud, there'd probably be a slaughter). She might be a rumor dreamt up by the authorities. After the Faction War, the only people who saw her were the heads of the various factions - what if they were lying, or misled?

I like to think that the Lady does a lot to protect Sigil that doesn't actually involve showing up personally. Maybe a dabus happens to be patching up a fence nearby to "keep an eye on things". Maybe a wild mage stumbles through a nearby portal and causes some comedic havoc to lessen tension.

LOL, like some kind of interplanar "Justice League of Amazing Superfriends"? That's actually a fairly interesting concept - a team of 20th level (or higher) cutters that can be assembled from across the planes on short notice to deal with threats above and beyond the local police force.

They have planar mancatchers to snare teleporters with, and probably a few high-level heavyweights. Basically, it has as much as you want it to have, but I wouldn't overdo it. Contra Union, you don't need epic-level characters on every street corner to handle a typical day on the beat, even in Sigil (or even in Union, for that matter). The typical citizen isn't that tough, after all.

If demons and mad archmages were continually trying to wreck the city just to wreck it, the police force would eventually be overwhelmed and the Lady of Pain would probably close all the portals, but that doesn't happen. Most powerful creatures who enter Sigil behave themselves, mostly; even demons have a decent respect for the law and the Lady, and the city is more useful to them intact.

I don't think the answer to the issue of powerful characters coming to the City of Doors is to add squads of 30th level legendary uberfighters with nothing better than to do than act as beat cops. I definitely don't think the answer is to have the Lady of Pain intervene personally every time a balor gets drunk. That degrades her and makes her less special and mysterious. The Planescape boxed set said she should only appear once or twice in a campaign, if that often, and I think that's about right. When you introduce a character who's nigh-omnipotent as the Lady of Pain is, the price that comes with that is you have to be very, very careful about overusing her. Otherwise she becomes a deus ex machina who the PCs think they can rely on to get them out of trouble, keep them from doing what they want, and generally keep excitement, drama, or anything of interest from happening in the campaign.

Instead, let the occasional abomination wreck havoc in the city, decimating a few streets until a few heroes (maybe including the PCs, or maybe they only hear about it), celestials, and whatever else manage to bring it down with the weight of sheer numbers. It happens; Sigil isn't a bastion of order in the multiverse, but a place of balance between Chaos and Law. Violence breaks out, not everything can be controlled, and the Lady of Pain either doesn't care, likes it that way, or lacks the power to act openly very often. Minor disasters happen continually, and the city police can't always cope with it all. The city remains just stable enough to function, and that's enough.

4th edition also makes the dabus pretty impressively powerful, many of them 18th level and no upper limit on how tough they can be. This is completely different from how it was handled in 2e, when they were all 1st-level minions who relied mostly on fear of the Lady and general inoffensiveness to survive. I have mixed feelings about that; I don't mind the idea that some dabus are more powerful than others, but I don't like the idea of them keeping the peace. It's not really their place. They're busy worker bees, not cops, and they shouldn't care any more about what mortals do in Sigil than the Lady of Pain does. So maybe there's an occasional dabus who can defend the others, but I think if a tarrasque started tearing down buildings they'd simply stare at it, wait for it to leave, and then start rebuilding, letting other people worry about subduing it. Sigil shouldn't be safe, and despite most of its inhabitants being low-level mortals, it isn't really built for mortals. They exist there on the Lady's sufferance; she doesn't owe them anything.

In summary:

1. Really powerful and dangerous characters are pretty rare.
2. Most of them behave themselves.
3. Those that don't may indeed cause some havoc before being defeated. That's okay. That's life in the center of the planes.
4. For truly catastrophic events, yes, the Lady of Pain can intervene... but sometimes she won't. If she always did, there'd be no sense that the city could ever be threatened, and there wouldn't be as much for the PCs to do. They shouldn't be allowed to feel too smug or safe... or to feel that their own mischief will inevitably fail.

In my view, the Lady of Pain shouldn't be used as an NPC. She shouldn't even be used as a plot device. She's background text, flavor, fluff, backstory, an origin myth, an enigma that reinforces that Sigil is a weird, mysterious place with a weirder, even more mysterious guardian.

As pure speculation on my part, I've always thought that, well, her name is "Lady of Pain." What if she wants people in Sigil to be having a crappy life? Not so crappy that they leave; just at the right level that a lot of people are miserable, but stay anyway. Thus, the Hive will never be successfully cleaned up, the in-city wars will always be brewing just under the surface, and questionably moral groups will always be serious voices in the governance of the city.

Would give you a bit of guideline on what level of pain she's willing to tolerate: the sustainable maximum.