As revealed at the end of The Final Empire, Steel Inquisitors have 11 metal spikes driven into various parts of their bodies. It would seem that there would be a relatively way for a Mistborn to kill the Inquisitor by Pulling on one spike while Pushing on another. I base my question on the fact that while normal Allomancers are unable to manipulate metals embedded in other people, the Lord Ruler could so it is theoretically possible. Would this be a valid method assuming the above?
They could be extracted with great Allomancy and it was intentional.
In later books (if I'm not wrong, in The Hero of Ages) it said that the Lord Ruler did that in order to keep the Inquisitors loyal to him. He designed them with Hemalurgy in a way that would make them vulnerable to his pulls/pushes. This way, they would live forever with the notion that their power came from him, and that he could kill them outright if they rebelled. I'm unable to find the exact passage, but maybe someone else can.
This is shown again when
Elend fights a steel Inquisitor in the Hero of Ages, and he discovers that the inquisitor had protected his central spike to avoid someone pulling on it (physically, at least).
This can be seen as Inquisitors trying to prevent the flaw they've been created with,
since at the time, with the Lord Ruler being dead, they weren't exactly "in check" anymore.
Generally, it's unlikely that anyone could push or pull on the spikes due to the nature of Allomancy. It is said that Allomantic powers have decreased over time (due to the power being passed down generation after generation, and "watered" down in the lineages) and we don't know if in the "golden era" of Allomancy it was possible.
The Lord Ruler was both an incredible Allomancer, and had a nice set of tricks up in his sleeve, so he could.
The Lord Ruler could do it. But it's unlikely that anyone else could
Mistborn is part of an overarching structured universe of Sanderson's called the Cosmere. One of the major rules of how magics interact in the Cosmere is that objects that have been charged with magic are very difficult to affect with magic, and this resistance scales with the amount of magic in them. It probably won't surprise you to learn that the spikes that power Inquisitors are very magical.
It is still possible to affect them with magic, but it requires a great deal more power than normal. Fortunately (or perhaps unfortunately) for the Lord Ruler, he is a great deal more powerful than the average Allomancer, and he would be able to achieve the feat.