The usual zooms you get with the k-x tend to be a little slow at the long end, especially indoors or in low light (I'm going to take a guess she has the DAL 18-55mm, and DAL 50-200mm ?)
The problem here is probably f-stop. The f-number is a ratio that indicates how much light is let in - the smaller the number, the better! It's a non linear scale, so f2.8 lets in a hell of a lot more light than f3.5. The zooms have a variable aperture, f3.5 at the wide end (18 or 50), and f5.6 at the long end (55 or 200). If she's zooming in a lot, the camera will need a slower shutter speed to get the correct exposure. This can lead to blur due to slight movement when the shutter is open. Technique can help, as can a tripod, but the easiest option is to get a lens with a bigger aperture!
So in theory, you're basically looking for a high quality zoom lens at longer focal lengths (for sports at any rate). The problem is that they are extremely expensive, and very heavy. (Eg 50-250 DA* f2.8)
As a result, I think you're going to need to look at prime lenses (ie lenses that only have one fixed focal length).
If she puts her photos on Flickr, it may be worth having a look at the info on the images to get an idea of what focal length she uses most often. Alternatively, turn on her camera, press the play button to view the images, and if you can't see picture details (eg 35mm f4.5), then press the info button until that appears
The two plastic fantastic primes (DA 50mm f1.8 and DA 35mm f2.4) are great value, much sharper, faster, and much better in low light. 35mm is a 'normal' length (similar to the human eye). 50mm is a bit narrower - usually used for head and shoulders portraits. Just those two alone will be an absolute revelation to her! (I'd say the 35mm is a no brainier - it's the most general purpose focal length - and therefore the one that will see most use)
Those two are probably a bit too wide for sporting events though
. Great all round lenses otherwise, and fairly cheap. Those are worth considering (she'll probably stop using the zooms altogether)
If she shoots a lot at zoomed in lengths (70+ mm), then maybe consider the DA 70mm f2.4 limited, or the DFA 100mm f2.8 WR Macro. The 70 is basically a fast-focussing portrait lens, and the macro will give her endless hour of fun taking pictures of insects and flowers. Macros aren't known for being the fastest focussing lenses, so that might not make it an ideal lens for sports, however the f2.8 f-stop will let in much more light than the zooms, allowing her to use faster shutter speeds. I'd prefer to shoot sports with that, compared to the 50-200.
All of those lenses are significantly sharper than her zooms, work better in lower light, and have good bokeh (the word used to describe the nice blurring you get in areas that are out of focus)
Whilst not really answering your exact question, I'd possibly go for the DA 35mm f2.4 (buy it second hand - they've recently been given away with cameras, so finding a more-or-less new one on eBay will be pretty easy). Everyone needs a good standard prime (it'll be perfect for general family photos). I'd pair that with either the 70 or the 100 (which are both metal with far higher build quality). The 100 adds macro capabilities, the 70 is faster focussing with a slightly bigger aperture - great for portraits.
Failing that, you can't go wrong with the two plastic fantastics (maybe upgrade the 50mm f1.8 to a DFA 50mm 2.8 macro? It's the cheapest of the macro lenses, and will give her hours of endless fun chasing insects)
Obligatory link to lens databases.
DA Zoom lenses:
https://www.pentaxforums.com/lensreviews/SMC-Pentax-DA-Zoom-Lenses-c43.html
DA primes:
https://www.pentaxforums.com/lensreviews/SMC-Pentax-DA-Lenses-c34.html
DFA primes:
https://www.pentaxforums.com/lensreviews/SMC-Pentax-DFA-Lenses-c33.html