The second period degenerated into a scrappy midfield tussle which was a disappointment after such an entertaining first half.
Harriers just could not get their act together and though they tightened up the defence no-one seemed to want to take responsibility for getting that vital third goal or even setting it up.
This draw, which will be very welcome in bottom of the table Swansea's relegation fight is bad news for Harriers' play-off hopes.
Though only Bury won in the Division Three top 10 so despite a disappointing display at least Harriers added another point to their tally.
However, It puts more pressure on Harriers impressive away from with trips to Lincoln and Macclesfield over the next week.
There was little in the way of chances in the second half watched by 3,172 fans.
Swansea midfielder Britton did fire one shot on target with Digby claiming comfortably by his post on 56 minutes.
Top scorer Henriksen was brought on 58 minutes to replace Broughton.
However, he went on to be cautioned 11 minutes from time for a foul on Hilton who had to be replaced by Howard due to injury.
Substitute Stamps was also cautioned for hand ball on 74 minutes, while Foster entered the fray on 81 minutes to replace Bishop.
Harriers man of the match Williams drove a low 20-yard free kick wide late on.
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