Mr.Wizard's answer correctly identifies why your code wasn't working: you needed StringPosition rather than Position, and then to delete the empty cases.
I wonder if there isn't another way to attack your problem. You are importing an HTML file, so why not use that fact rather than treat it as plain text? You don't actually need all the HTML markup. So instead consider:
ss = Import["http://www.swift.ac.uk/xrt_curves/00111529", "HTML"];
This strips out the extraneous HTML stuff. So the first 100 characters are:
StringTake[ss, 100]
"Skip banner and navigation tools .
Dept. of Physics & Astronomy
XROA
US Site | "
Contrast this with
s = Import["http://www.swift.ac.uk/xrt_curves/0000111529", "Text"];
StringTake[s, 100]

(I had to insert this as a picture because inserting an HTML DOCTYPE statement really doesn't seem to work nicely with the StackExchange software, and rightly so!)
As you can see, you can obtain the positions of the desired substring within the whole, without having to split the string first.
StringPosition[ss, "Flux Light Curve"]
{{1797, 1812}, {2435, 2450}, {2465, 2480}}
Of course, you can always StringSplit in the same way and find the substring that contains the sub(sub)string:
sss = StringSplit[ss, "\n"];
StringPosition[sss, "Flux Light Curve"]
{{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {{52, 67}},
{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
{}, {}, {{1, 16}, {31, 46}}, {}, {}, {}, {}, {}, {}}
Position[{"ab", "bc", "ac", "be"}, x_ /; StringMatchQ[x, "b" ~~ __]]>>StringMatchQ::strse: String or list of strings expected at position 1 in StringMatchQ[List,b~~__].– alancalvitti Jul 21 '17 at 18:53Positionlooks at all parts of the expression, including heads, in this caseList. See (7692), (67100). Simple solution: only pass expressions that match_StringtoStringMatchQ:Position[{"ab", "bc", "ac", "be"}, x_String /; StringMatchQ[x, "b" ~~ __]]—this is more general than addingHeads -> False. – Mr.Wizard Jul 22 '17 at 00:18