We study polarization-dependent spontaneous emission (SE) rate and light extraction efficiency (LEE) in localized-surface-plasmon (LSP)-coupled light emitting diodes (LEDs). The closely packed seven $Ag/SiO_2$ core-shell (CS) nanoparticles (NPs) lie on top of the GaN surface for LSP coupling with a radiated dipole. According to the dipole direction, both the SE rate and the LEE are significantly modified by the LSP effect at the $Ag/SiO_2$ CS NPs when the size of Ag, the thickness of $SiO_2$, and the position of the dipole source are varied. The enhancement of the SE rate is related to an induced dipole effect at the Ag, and the high LEE is caused by light scattering with an LSP mode at $Ag/SiO_2$ CS NPs. We suggest the optimum position of the quantum well (QW) in blue InGaN/GaN LEDs with $Ag/SiO_2$ CS NPs for practical application.