Your speakers are fine for the karaoke music but I don't recommend singing through them.
Most people are not audio experts, they just want to grab a mic and sing. If they don't understand concepts like stage gaining you could damage your AV speakers if you try to sing through them.
I recommend a mixer and a powered speaker to sing through.
Because the powered speakers I recommend they already have a simple mixer so you don't absolutely need a separate mixer. But I recommend one because they have vocals effects and it's more convenient having the controls close to you instead of reaching around the back of the speaker all the time.
Mixer
I don't own this mixer but it's highly recommended by others. I recommend it because you can use standard XLR cables to connect it to most powered speakers.
https://www.amazon.com/Yamaha-MG06X-6-Input-Compact-Effects/dp/B00I2J4V5G?th=1
Speaker
https://www.amazon.com/Mackie-Thump12A-1300W-Powered-Loudspeaker/dp/B073V7MKQZ
https://smile.amazon.com/Rockville-RPG10BT-V2-Bluetooth-Wireless/dp/B084KZCY54?th=1
If you want to encourage her to sing and record her songs then option 2 is the way to go.
For a powered speaker I usually recommend this 12"Rockville that has wheels. I have one and it's decent for home karaoke.
https://www.amazon.com/Rockville-RPG15BT-V2-Bluetooth-Additional/dp/B084KZCY54
I talked about it here
https://www.reddit.com/r/karaoke/comments/pa25ud/what_is_the_best_allinone_mixerpower_ampspeaker/
and this looked pretty nice at Sam's Club
https://www.samsclub.com/p/ion-pa-live-bluetooth-speaker/prod25480027?xid=plp_product_1
But if money is no object then you might want to check out this discussion
https://www.reddit.com/r/karaoke/comments/pn3kut/the_best_powered_speaker_for_home_karaoke_when/
If you try to use bluetooth for karaoke you'll have a noticeable delay between the lyrics and the music. You might decide it's good enough but you'll get a better experience if you connect the audio from the phone directly to the PA.
There are all-in-one powered speakers with mics. They might work for you but I don't recommend them because the mics aren't very good quality.
https://smile.amazon.com/Rockville-Rechargable-Bluetooth-RAM15BT-V2/dp/B07P2DW1VY?th=1
The other potential drawback to this system is the controls are on the back of the speaker. Depending on the setup it can be difficult to reach the controls.
Here is my post for a cheap home setup.
The best quality system in your price range would be this for the powered speaker
https://www.amazon.com/gp/product/B084KZCY54?th=1
and these mics
https://www.amazon.com/innopow-Microphone-150-200Ft-Continuous-WM-200-New/dp/B07F58YP2Z