Can I use this piece (see pic) as a replacement?
It looks like you have two female spade connectors and a female-to-female connector, and that's the part that broke.  The female-to-two-stacked-male connector you have pictured may work, you just wouldn't be using the female part of it.  

I find that there are small differences between brands and things don't mate as well as they should.  If the wires are in good shape and not too short, a lot of times the easiest thing is to just cut off what's there and replace with your brand.  

I don't see any compelling reason to use two female connectors here - I might replace these with one female and one male so you don't have the connector to deal with.  I'd also use the fully insulated connectors, just to be a little safer, less chance of accidental contact, shorts, etc., especially since it's line voltage.  

You can connect these wires by any suitable means.  Crimp on connectors are easy to disconnect / reconnect without tools but you could use a wire nut, lever nut, etc. - anything listed and rated for the voltage and current.  

Keep in mind that a lot of crimp terminal products are NOT listed / rated for line voltage, aimed more at the low voltage of the automotive market.  I stick to reputable manufacturers' brands.