I'm not going to get into yet another massive argument about how I'm doing it wrong, but I'll just leave a few points here...
The situation is, from Taylor's point of view:
- She and her family and friends came under fire from an unknown enemy and legged it using their evacuation plan. In her case, this went wrong.
- When she worked out that it had gone wrong, she initially and fairly reasonably assumed that it had only gone a little wrong and she was in roughly the right place but in the wrong exact location.
- Based on that, she headed toward where she thought she should have ended up rather than away.
- It would also have been much trickier to exit the vent system on the surface, for reasons of distance, intervening machinery, and probably security. And as in the point above, at that point she had no reason to really believe her actions weren't correct.
- By the time she realized that she was a lot further away from home than she'd expected, she was also fully aware she was in an unknown and almost certainly top secret military base.
- Unknown people popping up in top secret military bases tend to upset the military running those bases. She's not stupid, she knows that full well.
- As a result of this, simply dropping out of a vent and saying brightly "Hi, there, I'm Taylor and I'm in the wrong world! Can I borrow your Stargate?" is exceptionally unlikely to work out well for her or anyone else. Again, she's easily able to work this out without trying it

- Getting captured unwillingly by this unknown military is even less likely to work out well. In ways that range from being shot on sight to put in jail forever. So that's also a thing to avoid.
- While the SG people are in reality somewhat more reasonable than she might fear, she doesn't know that and can't take the chance. And even if General Hammond was willing to help, he'd be obliged to report it upstairs and those people would definitely immediately decide that she and all her toys were now their property. They've got form for that sort of thing...
- There's enough stuff lying around she can borrow to build a way home, especially once she's studied the situation.
- There's a place that will give enough privacy for long enough (nearly) to do this unobserved. Another hour or so and she'd have been gone by the time they walked through the door.
- Conveniently there's also equipment around the place that makes her own security easier to arrange. But if that hadn't been present she'd have done something else.
- She 'paid' for the inconvenience of temporarily misplacing alien technology that the SG people did, after all, either simply stumble across or flat out steal by fixing some of it, making notes on much of the rest, and leaving a document behind on purpose that would jump them decades ahead in one step. It's a fair deal even if not according to normal military protocol. Sam would vehemently agree...
- By the time they found her, she'd been working for about 20 hours straight without sleep, was very worried about her dad and friends, was scared about what might happen if she didn't get back, and was basically a little punch-drunk from exhaustion and stress. As a result she was slightly more talkative than normal

So, given the above, what else was she reasonably expected to do? Throwing herself on the mercy of people who would almost certainly take her appearing out of nowhere inside an obviously very secret place was about the last thing she'd try. Escaping entirely from the base was probably possible, but by the time she was sure that she might have to, it would have been much more difficult and even if she succeeded would put most of the stuff she required for a quick exit out of reach. Again, not something to do other than a last resort.
She did the best she could with what she had and what she knew of the situation. I can't really see what other choices she had.
And of course this is an omake and just for fun, sparked by a comment made by a reader. So you don't have to take it seriously if you don't want to