Is there any relation between LJU (Line Jack Unit) and broadband speed?
Yes, there seems to be a lot of evidence that a proper NTU can affect the broadband speed.

I don't know what the rules are in India, but here in the UK it's BT's responsibility to provide and maintain the NTU that terminates the incoming phone line and consumers aren't allowed to interfere with it. The best results are reckoned to come from having a properly filtered faceplate as part of the NTU because this removes the effect of the telephone wiring as close to the entry point as possible. But in your case I wouldn't really expect a massive improvement in speed, if any, given the short length to the splitter.

But broadband speed is mainly dependent on distance to the DSLAM (either at the exchange or in a street cabinet) and the quality of cabling. Prior to rolling out street cabinets to make shorter connections, it was quite normal to get 6Mbps from broadband - and many people still can't even get that much. Have you asked your provider what speed you should be getting?