With version 9.0.1,
f[x_] := (p^2 + k^2 - 2 p k x)/(x - (p^2 + k^2 + 1 - ((p^2 - k^2)^2)/4)/(2 p k));
ans9 = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True]
(* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2)
ArcCoth[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)),
k^4 + p^4 < 4 + 4 p^2 + 2 k^2 (2 + p^2) && (k - p)^2 (-2 + k + p) (2 + k + p) > 4] *)
Some insight can be gained by plotting the solution ad its region of validity as specified by ConditionalExpression,
RegionPlot[Evaluate[ans9[[2]]], {k, -5, 5}, {p, -5, 5}, FrameLabel -> {k, p}]
Plot3D[Evaluate[ans9[[1]]], {k, -5, 5}, {p, -5, 5}, PlotPoints -> 100,
PlotRange -> All, Mesh -> None, AxesLabel -> {k, p, z}]
Thus, the solution is real and continuous over a wider range than that given by the ConditionalExpression. Nonetheless, the actual range is bounded by singularities, so it is not surprising that the second Integrate in the question does not yield an answer. Of course, even without the singularities, Integrate might fail, if no known symbolic solution exists.
Because, I presume, k and p are meant to be Reals, it is reasonable to inform Integrate of this.
ans9r = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True,
Assumptions -> k ∈ Reals && p ∈ Reals]
(* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2)
ArcTanh[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)),
(-4 + (k - p)^2) (k + p)^2 <= 4 && (k - p)^2 (-2 + k + p) (2 + k + p) <= 4] *)
which produces a different but related solution (perhaps, a different branch).
Between them, the two solutions appear to cover all of p - k space. Nonetheless, it seems unlikely that patching the two together and then integrating would be successful.
Version 10.1 produces symbolic solutions that look different.
(* ConditionalExpression[1/4 (-(-4 + (k^2 - p^2)^2) Log[-k p] + (k^2 - p^2)^2 (Log[k p]
- Log[4 - (-4 + (k - p)^2) (k + p)^2] + Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])
- 4 (4 k p + Log[k p] - Log[4 - (-4 + (k - p)^2) (k + p)^2] +
Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])),
(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))/(k p) ∈ Reals] *)
and
(* ConditionalExpression[1/4 (-16 k p - (-4 + (k^2 - p^2)^2)
Log[4 - (-4 + (k - p)^2) (k + p)^2] + (-4 + (k^2 - p^2)^2)
Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)]),
k^4 + p^4 <= 4 + 4 p^2 + 2 k^2 (2 + p^2) && (-4 + (k - p)^2) (k + p)^2 <= 4] *)
Their 3D plots appear identical respectively the fourth and second plots above, although their asserted ranges of validity differ. Note that a straightforward application of FullSimplify does not succeed in showing them to be identical.